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Theorem sticksstones1 41054
Description: Different strictly monotone functions have different ranges. (Contributed by metakunt, 27-Sep-2024.)
Hypotheses
Ref Expression
sticksstones1.1 (𝜑𝑁 ∈ ℕ0)
sticksstones1.2 (𝜑𝐾 ∈ ℕ0)
sticksstones1.3 𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))}
sticksstones1.4 (𝜑𝑋𝐴)
sticksstones1.5 (𝜑𝑌𝐴)
sticksstones1.6 (𝜑𝑋𝑌)
sticksstones1.7 𝐼 = inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < )
Assertion
Ref Expression
sticksstones1 (𝜑 → ran 𝑋 ≠ ran 𝑌)
Distinct variable groups:   𝐴,𝑓   𝑥,𝐼,𝑦   𝑧,𝐼   𝑓,𝐾,𝑥,𝑦   𝑧,𝐾   𝑓,𝑁   𝑓,𝑋,𝑥,𝑦   𝑧,𝑋   𝑓,𝑌,𝑥,𝑦   𝑧,𝑌   𝜑,𝑓
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥,𝑦,𝑧)   𝐼(𝑓)   𝑁(𝑥,𝑦,𝑧)

Proof of Theorem sticksstones1
Dummy variables 𝑗 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sticksstones1.7 . . . . . 6 𝐼 = inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < )
21a1i 11 . . . . 5 (𝜑𝐼 = inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < ))
3 ltso 11296 . . . . . . 7 < Or ℝ
43a1i 11 . . . . . 6 (𝜑 → < Or ℝ)
5 fzfid 13940 . . . . . . . 8 (𝜑 → (1...𝐾) ∈ Fin)
6 ssrab2 4077 . . . . . . . . 9 {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ⊆ (1...𝐾)
76a1i 11 . . . . . . . 8 (𝜑 → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ⊆ (1...𝐾))
8 ssfi 9175 . . . . . . . 8 (((1...𝐾) ∈ Fin ∧ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ⊆ (1...𝐾)) → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ∈ Fin)
95, 7, 8syl2anc 584 . . . . . . 7 (𝜑 → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ∈ Fin)
10 sticksstones1.6 . . . . . . . 8 (𝜑𝑋𝑌)
11 rabeq0 4384 . . . . . . . . . . . . 13 ({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} = ∅ ↔ ∀𝑧 ∈ (1...𝐾) ¬ (𝑋𝑧) ≠ (𝑌𝑧))
12 nne 2944 . . . . . . . . . . . . . 14 (¬ (𝑋𝑧) ≠ (𝑌𝑧) ↔ (𝑋𝑧) = (𝑌𝑧))
1312ralbii 3093 . . . . . . . . . . . . 13 (∀𝑧 ∈ (1...𝐾) ¬ (𝑋𝑧) ≠ (𝑌𝑧) ↔ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧))
1411, 13bitri 274 . . . . . . . . . . . 12 ({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} = ∅ ↔ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧))
15 feq1 6698 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑋 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑋:(1...𝐾)⟶(1...𝑁)))
16 fveq1 6890 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 = 𝑋 → (𝑓𝑥) = (𝑋𝑥))
17 fveq1 6890 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 = 𝑋 → (𝑓𝑦) = (𝑋𝑦))
1816, 17breq12d 5161 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑋 → ((𝑓𝑥) < (𝑓𝑦) ↔ (𝑋𝑥) < (𝑋𝑦)))
1918imbi2d 340 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = 𝑋 → ((𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ (𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦))))
20192ralbidv 3218 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑋 → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦))))
2115, 20anbi12d 631 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑋 → ((𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦))) ↔ (𝑋:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)))))
22 sticksstones1.3 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))}
23 eqabb 2873 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))} ↔ ∀𝑓(𝑓𝐴 ↔ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))))
2422, 23mpbi 229 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑓(𝑓𝐴 ↔ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦))))
2524spi 2177 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓𝐴 ↔ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦))))
2625biimpi 215 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓𝐴 → (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦))))
2726adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑓𝐴) → (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦))))
2827ralrimiva 3146 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ∀𝑓𝐴 (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦))))
29 sticksstones1.4 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑋𝐴)
3021, 28, 29rspcdva 3613 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑋:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦))))
3130simpld 495 . . . . . . . . . . . . . . . . . 18 (𝜑𝑋:(1...𝐾)⟶(1...𝑁))
3231ffnd 6718 . . . . . . . . . . . . . . . . 17 (𝜑𝑋 Fn (1...𝐾))
3332adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧)) → 𝑋 Fn (1...𝐾))
34 sticksstones1.5 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑌𝐴)
35 feq1 6698 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑌 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑌:(1...𝐾)⟶(1...𝑁)))
36 fveq1 6890 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = 𝑌 → (𝑓𝑥) = (𝑌𝑥))
37 fveq1 6890 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = 𝑌 → (𝑓𝑦) = (𝑌𝑦))
3836, 37breq12d 5161 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 = 𝑌 → ((𝑓𝑥) < (𝑓𝑦) ↔ (𝑌𝑥) < (𝑌𝑦)))
3938imbi2d 340 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 = 𝑌 → ((𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ (𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦))))
40392ralbidv 3218 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑌 → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦))))
4135, 40anbi12d 631 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = 𝑌 → ((𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦))) ↔ (𝑌:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)))))
4241, 28, 34rspcdva 3613 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑌:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦))))
4342adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑌𝐴) → (𝑌:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦))))
4434, 43mpdan 685 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑌:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦))))
4544simpld 495 . . . . . . . . . . . . . . . . . 18 (𝜑𝑌:(1...𝐾)⟶(1...𝑁))
4645ffnd 6718 . . . . . . . . . . . . . . . . 17 (𝜑𝑌 Fn (1...𝐾))
4746adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧)) → 𝑌 Fn (1...𝐾))
48 eqfnfv 7032 . . . . . . . . . . . . . . . 16 ((𝑋 Fn (1...𝐾) ∧ 𝑌 Fn (1...𝐾)) → (𝑋 = 𝑌 ↔ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧)))
4933, 47, 48syl2anc 584 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧)) → (𝑋 = 𝑌 ↔ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧)))
5049bicomd 222 . . . . . . . . . . . . . 14 ((𝜑 ∧ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧)) → (∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧) ↔ 𝑋 = 𝑌))
5150biimpd 228 . . . . . . . . . . . . 13 ((𝜑 ∧ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧)) → (∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧) → 𝑋 = 𝑌))
5251syldbl2 839 . . . . . . . . . . . 12 ((𝜑 ∧ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧)) → 𝑋 = 𝑌)
5314, 52sylan2b 594 . . . . . . . . . . 11 ((𝜑 ∧ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} = ∅) → 𝑋 = 𝑌)
5453ex 413 . . . . . . . . . 10 (𝜑 → ({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} = ∅ → 𝑋 = 𝑌))
5554necon3d 2961 . . . . . . . . 9 (𝜑 → (𝑋𝑌 → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ≠ ∅))
5655imp 407 . . . . . . . 8 ((𝜑𝑋𝑌) → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ≠ ∅)
5710, 56mpdan 685 . . . . . . 7 (𝜑 → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ≠ ∅)
58 fz1ssnn 13534 . . . . . . . . . 10 (1...𝐾) ⊆ ℕ
5958a1i 11 . . . . . . . . 9 (𝜑 → (1...𝐾) ⊆ ℕ)
60 nnssre 12218 . . . . . . . . . 10 ℕ ⊆ ℝ
6160a1i 11 . . . . . . . . 9 (𝜑 → ℕ ⊆ ℝ)
6259, 61sstrd 3992 . . . . . . . 8 (𝜑 → (1...𝐾) ⊆ ℝ)
637, 62sstrd 3992 . . . . . . 7 (𝜑 → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ⊆ ℝ)
649, 57, 633jca 1128 . . . . . 6 (𝜑 → ({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ∈ Fin ∧ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ≠ ∅ ∧ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ⊆ ℝ))
65 fiinfcl 9498 . . . . . 6 (( < Or ℝ ∧ ({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ∈ Fin ∧ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ≠ ∅ ∧ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ⊆ ℝ)) → inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < ) ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)})
664, 64, 65syl2anc 584 . . . . 5 (𝜑 → inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < ) ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)})
672, 66eqeltrd 2833 . . . 4 (𝜑𝐼 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)})
687, 66sseldd 3983 . . . . . 6 (𝜑 → inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < ) ∈ (1...𝐾))
692eleq1d 2818 . . . . . 6 (𝜑 → (𝐼 ∈ (1...𝐾) ↔ inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < ) ∈ (1...𝐾)))
7068, 69mpbird 256 . . . . 5 (𝜑𝐼 ∈ (1...𝐾))
71 fveq2 6891 . . . . . . 7 (𝑧 = 𝐼 → (𝑋𝑧) = (𝑋𝐼))
72 fveq2 6891 . . . . . . 7 (𝑧 = 𝐼 → (𝑌𝑧) = (𝑌𝐼))
7371, 72neeq12d 3002 . . . . . 6 (𝑧 = 𝐼 → ((𝑋𝑧) ≠ (𝑌𝑧) ↔ (𝑋𝐼) ≠ (𝑌𝐼)))
7473elrab3 3684 . . . . 5 (𝐼 ∈ (1...𝐾) → (𝐼 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ↔ (𝑋𝐼) ≠ (𝑌𝐼)))
7570, 74syl 17 . . . 4 (𝜑 → (𝐼 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ↔ (𝑋𝐼) ≠ (𝑌𝐼)))
7667, 75mpbid 231 . . 3 (𝜑 → (𝑋𝐼) ≠ (𝑌𝐼))
77 nfv 1917 . . . . . 6 𝑎𝜑
78 nfcv 2903 . . . . . 6 𝑎(1...𝑁)
79 nfcv 2903 . . . . . 6 𝑎
80 elfznn 13532 . . . . . . . . 9 (𝑎 ∈ (1...𝑁) → 𝑎 ∈ ℕ)
8180adantl 482 . . . . . . . 8 ((𝜑𝑎 ∈ (1...𝑁)) → 𝑎 ∈ ℕ)
82 nnre 12221 . . . . . . . 8 (𝑎 ∈ ℕ → 𝑎 ∈ ℝ)
8381, 82syl 17 . . . . . . 7 ((𝜑𝑎 ∈ (1...𝑁)) → 𝑎 ∈ ℝ)
8483ex 413 . . . . . 6 (𝜑 → (𝑎 ∈ (1...𝑁) → 𝑎 ∈ ℝ))
8577, 78, 79, 84ssrd 3987 . . . . 5 (𝜑 → (1...𝑁) ⊆ ℝ)
8631, 70ffvelcdmd 7087 . . . . 5 (𝜑 → (𝑋𝐼) ∈ (1...𝑁))
8785, 86sseldd 3983 . . . 4 (𝜑 → (𝑋𝐼) ∈ ℝ)
8845, 70ffvelcdmd 7087 . . . . 5 (𝜑 → (𝑌𝐼) ∈ (1...𝑁))
8985, 88sseldd 3983 . . . 4 (𝜑 → (𝑌𝐼) ∈ ℝ)
90 lttri2 11298 . . . 4 (((𝑋𝐼) ∈ ℝ ∧ (𝑌𝐼) ∈ ℝ) → ((𝑋𝐼) ≠ (𝑌𝐼) ↔ ((𝑋𝐼) < (𝑌𝐼) ∨ (𝑌𝐼) < (𝑋𝐼))))
9187, 89, 90syl2anc 584 . . 3 (𝜑 → ((𝑋𝐼) ≠ (𝑌𝐼) ↔ ((𝑋𝐼) < (𝑌𝐼) ∨ (𝑌𝐼) < (𝑋𝐼))))
9276, 91mpbid 231 . 2 (𝜑 → ((𝑋𝐼) < (𝑌𝐼) ∨ (𝑌𝐼) < (𝑋𝐼)))
9331ffund 6721 . . . . . 6 (𝜑 → Fun 𝑋)
9493adantr 481 . . . . 5 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → Fun 𝑋)
9531fdmd 6728 . . . . . . 7 (𝜑 → dom 𝑋 = (1...𝐾))
9670, 95eleqtrrd 2836 . . . . . 6 (𝜑𝐼 ∈ dom 𝑋)
9796adantr 481 . . . . 5 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → 𝐼 ∈ dom 𝑋)
98 fvelrn 7078 . . . . 5 ((Fun 𝑋𝐼 ∈ dom 𝑋) → (𝑋𝐼) ∈ ran 𝑋)
9994, 97, 98syl2anc 584 . . . 4 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → (𝑋𝐼) ∈ ran 𝑋)
100 elfznn 13532 . . . . . . . . . . . 12 (𝑗 ∈ (1...𝐾) → 𝑗 ∈ ℕ)
1011003ad2ant3 1135 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝑗 ∈ ℕ)
102101nnred 12229 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝑗 ∈ ℝ)
10362, 70sseldd 3983 . . . . . . . . . . 11 (𝜑𝐼 ∈ ℝ)
1041033ad2ant1 1133 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝐼 ∈ ℝ)
105102, 104lttri4d 11357 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑗 < 𝐼𝑗 = 𝐼𝐼 < 𝑗))
106453ad2ant1 1133 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝑌:(1...𝐾)⟶(1...𝑁))
107 simp3 1138 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝑗 ∈ (1...𝐾))
108106, 107ffvelcdmd 7087 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑌𝑗) ∈ (1...𝑁))
109 fz1ssnn 13534 . . . . . . . . . . . . . . 15 (1...𝑁) ⊆ ℕ
110109sseli 3978 . . . . . . . . . . . . . 14 ((𝑌𝑗) ∈ (1...𝑁) → (𝑌𝑗) ∈ ℕ)
111 nnre 12221 . . . . . . . . . . . . . 14 ((𝑌𝑗) ∈ ℕ → (𝑌𝑗) ∈ ℝ)
112110, 111syl 17 . . . . . . . . . . . . 13 ((𝑌𝑗) ∈ (1...𝑁) → (𝑌𝑗) ∈ ℝ)
113108, 112syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑌𝑗) ∈ ℝ)
114113adantr 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑌𝑗) ∈ ℝ)
11530simprd 496 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)))
1161153ad2ant1 1133 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)))
117116adantr 481 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)))
118 simpl3 1193 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → 𝑗 ∈ (1...𝐾))
119703ad2ant1 1133 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝐼 ∈ (1...𝐾))
120119adantr 481 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → 𝐼 ∈ (1...𝐾))
121 breq1 5151 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑗 → (𝑥 < 𝑦𝑗 < 𝑦))
122 fveq2 6891 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑗 → (𝑋𝑥) = (𝑋𝑗))
123122breq1d 5158 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑗 → ((𝑋𝑥) < (𝑋𝑦) ↔ (𝑋𝑗) < (𝑋𝑦)))
124121, 123imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑗 → ((𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)) ↔ (𝑗 < 𝑦 → (𝑋𝑗) < (𝑋𝑦))))
125 breq2 5152 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐼 → (𝑗 < 𝑦𝑗 < 𝐼))
126 fveq2 6891 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝐼 → (𝑋𝑦) = (𝑋𝐼))
127126breq2d 5160 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐼 → ((𝑋𝑗) < (𝑋𝑦) ↔ (𝑋𝑗) < (𝑋𝐼)))
128125, 127imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐼 → ((𝑗 < 𝑦 → (𝑋𝑗) < (𝑋𝑦)) ↔ (𝑗 < 𝐼 → (𝑋𝑗) < (𝑋𝐼))))
129124, 128rspc2v 3622 . . . . . . . . . . . . . . 15 ((𝑗 ∈ (1...𝐾) ∧ 𝐼 ∈ (1...𝐾)) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)) → (𝑗 < 𝐼 → (𝑋𝑗) < (𝑋𝐼))))
130118, 120, 129syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)) → (𝑗 < 𝐼 → (𝑋𝑗) < (𝑋𝐼))))
131117, 130mpd 15 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑗 < 𝐼 → (𝑋𝑗) < (𝑋𝐼)))
132131syldbl2 839 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑋𝑗) < (𝑋𝐼))
133 simp2 1137 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → 𝑗 ∈ (1...𝐾))
134 simp3 1138 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → 𝑗 < 𝐼)
1351003ad2ant2 1134 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → 𝑗 ∈ ℕ)
136135nnred 12229 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → 𝑗 ∈ ℝ)
1371033ad2ant1 1133 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → 𝐼 ∈ ℝ)
138136, 137ltnled 11363 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → (𝑗 < 𝐼 ↔ ¬ 𝐼𝑗))
139134, 138mpbid 231 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → ¬ 𝐼𝑗)
140633ad2ant1 1133 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ⊆ ℝ)
14193ad2ant1 1133 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ∈ Fin)
142 infrefilb 12202 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ⊆ ℝ ∧ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ∈ Fin ∧ 𝑗 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}) → inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < ) ≤ 𝑗)
1431423expia 1121 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ⊆ ℝ ∧ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ∈ Fin) → (𝑗 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} → inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < ) ≤ 𝑗))
144140, 141, 143syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → (𝑗 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} → inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < ) ≤ 𝑗))
145144imp 407 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) ∧ 𝑗 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}) → inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < ) ≤ 𝑗)
1461a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) ∧ 𝑗 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}) → 𝐼 = inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < ))
147146breq1d 5158 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) ∧ 𝑗 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}) → (𝐼𝑗 ↔ inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < ) ≤ 𝑗))
148145, 147mpbird 256 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) ∧ 𝑗 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}) → 𝐼𝑗)
149148ex 413 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → (𝑗 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} → 𝐼𝑗))
150149con3d 152 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → (¬ 𝐼𝑗 → ¬ 𝑗 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}))
151139, 150mpd 15 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → ¬ 𝑗 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)})
152 nfcv 2903 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑧𝑗
153 nfcv 2903 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑧(1...𝐾)
154 nfv 1917 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑧(𝑋𝑗) ≠ (𝑌𝑗)
155 fveq2 6891 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = 𝑗 → (𝑋𝑧) = (𝑋𝑗))
156 fveq2 6891 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = 𝑗 → (𝑌𝑧) = (𝑌𝑗))
157155, 156neeq12d 3002 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 = 𝑗 → ((𝑋𝑧) ≠ (𝑌𝑧) ↔ (𝑋𝑗) ≠ (𝑌𝑗)))
158152, 153, 154, 157elrabf 3679 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ↔ (𝑗 ∈ (1...𝐾) ∧ (𝑋𝑗) ≠ (𝑌𝑗)))
159158notbii 319 . . . . . . . . . . . . . . . . . . . . . 22 𝑗 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ↔ ¬ (𝑗 ∈ (1...𝐾) ∧ (𝑋𝑗) ≠ (𝑌𝑗)))
160 ianor 980 . . . . . . . . . . . . . . . . . . . . . 22 (¬ (𝑗 ∈ (1...𝐾) ∧ (𝑋𝑗) ≠ (𝑌𝑗)) ↔ (¬ 𝑗 ∈ (1...𝐾) ∨ ¬ (𝑋𝑗) ≠ (𝑌𝑗)))
161159, 160bitri 274 . . . . . . . . . . . . . . . . . . . . 21 𝑗 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ↔ (¬ 𝑗 ∈ (1...𝐾) ∨ ¬ (𝑋𝑗) ≠ (𝑌𝑗)))
162151, 161sylib 217 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → (¬ 𝑗 ∈ (1...𝐾) ∨ ¬ (𝑋𝑗) ≠ (𝑌𝑗)))
163 imor 851 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ (1...𝐾) → ¬ (𝑋𝑗) ≠ (𝑌𝑗)) ↔ (¬ 𝑗 ∈ (1...𝐾) ∨ ¬ (𝑋𝑗) ≠ (𝑌𝑗)))
164162, 163sylibr 233 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → (𝑗 ∈ (1...𝐾) → ¬ (𝑋𝑗) ≠ (𝑌𝑗)))
165164imp 407 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) ∧ 𝑗 ∈ (1...𝐾)) → ¬ (𝑋𝑗) ≠ (𝑌𝑗))
166 nne 2944 . . . . . . . . . . . . . . . . . 18 (¬ (𝑋𝑗) ≠ (𝑌𝑗) ↔ (𝑋𝑗) = (𝑌𝑗))
167165, 166sylib 217 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝑗) = (𝑌𝑗))
168133, 167mpdan 685 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → (𝑋𝑗) = (𝑌𝑗))
1691683expa 1118 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑋𝑗) = (𝑌𝑗))
1701693adantl2 1167 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑋𝑗) = (𝑌𝑗))
171170eqcomd 2738 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑌𝑗) = (𝑋𝑗))
172171breq1d 5158 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → ((𝑌𝑗) < (𝑋𝐼) ↔ (𝑋𝑗) < (𝑋𝐼)))
173132, 172mpbird 256 . . . . . . . . . . 11 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑌𝑗) < (𝑋𝐼))
174114, 173ltned 11352 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑌𝑗) ≠ (𝑋𝐼))
175763ad2ant1 1133 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝐼) ≠ (𝑌𝐼))
176175adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑋𝐼) ≠ (𝑌𝐼))
177176necomd 2996 . . . . . . . . . . 11 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑌𝐼) ≠ (𝑋𝐼))
178 fveq2 6891 . . . . . . . . . . . . 13 (𝑗 = 𝐼 → (𝑌𝑗) = (𝑌𝐼))
179178neeq1d 3000 . . . . . . . . . . . 12 (𝑗 = 𝐼 → ((𝑌𝑗) ≠ (𝑋𝐼) ↔ (𝑌𝐼) ≠ (𝑋𝐼)))
180179adantl 482 . . . . . . . . . . 11 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → ((𝑌𝑗) ≠ (𝑋𝐼) ↔ (𝑌𝐼) ≠ (𝑋𝐼)))
181177, 180mpbird 256 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑌𝑗) ≠ (𝑋𝐼))
182873ad2ant1 1133 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝐼) ∈ ℝ)
183182adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝐼) ∈ ℝ)
184893ad2ant1 1133 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑌𝐼) ∈ ℝ)
185184adantr 481 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝐼) ∈ ℝ)
186113adantr 481 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝑗) ∈ ℝ)
187 simpl2 1192 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝐼) < (𝑌𝐼))
18842simprd 496 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)))
1891883ad2ant1 1133 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)))
190189adantr 481 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)))
191119adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → 𝐼 ∈ (1...𝐾))
192107adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → 𝑗 ∈ (1...𝐾))
193 breq1 5151 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐼 → (𝑥 < 𝑦𝐼 < 𝑦))
194 fveq2 6891 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝐼 → (𝑌𝑥) = (𝑌𝐼))
195194breq1d 5158 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐼 → ((𝑌𝑥) < (𝑌𝑦) ↔ (𝑌𝐼) < (𝑌𝑦)))
196193, 195imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐼 → ((𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)) ↔ (𝐼 < 𝑦 → (𝑌𝐼) < (𝑌𝑦))))
197 breq2 5152 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑗 → (𝐼 < 𝑦𝐼 < 𝑗))
198 fveq2 6891 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑗 → (𝑌𝑦) = (𝑌𝑗))
199198breq2d 5160 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑗 → ((𝑌𝐼) < (𝑌𝑦) ↔ (𝑌𝐼) < (𝑌𝑗)))
200197, 199imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑗 → ((𝐼 < 𝑦 → (𝑌𝐼) < (𝑌𝑦)) ↔ (𝐼 < 𝑗 → (𝑌𝐼) < (𝑌𝑗))))
201196, 200rspc2v 3622 . . . . . . . . . . . . . . . 16 ((𝐼 ∈ (1...𝐾) ∧ 𝑗 ∈ (1...𝐾)) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)) → (𝐼 < 𝑗 → (𝑌𝐼) < (𝑌𝑗))))
202191, 192, 201syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)) → (𝐼 < 𝑗 → (𝑌𝐼) < (𝑌𝑗))))
203190, 202mpd 15 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝐼 < 𝑗 → (𝑌𝐼) < (𝑌𝑗)))
204203syldbl2 839 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝐼) < (𝑌𝑗))
205183, 185, 186, 187, 204lttrd 11377 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝐼) < (𝑌𝑗))
206183, 205ltned 11352 . . . . . . . . . . 11 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝐼) ≠ (𝑌𝑗))
207206necomd 2996 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝑗) ≠ (𝑋𝐼))
208174, 181, 2073jaodan 1430 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ (𝑗 < 𝐼𝑗 = 𝐼𝐼 < 𝑗)) → (𝑌𝑗) ≠ (𝑋𝐼))
209105, 208mpdan 685 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑌𝑗) ≠ (𝑋𝐼))
2102093expa 1118 . . . . . . 7 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) ∧ 𝑗 ∈ (1...𝐾)) → (𝑌𝑗) ≠ (𝑋𝐼))
211210neneqd 2945 . . . . . 6 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) ∧ 𝑗 ∈ (1...𝐾)) → ¬ (𝑌𝑗) = (𝑋𝐼))
212211ralrimiva 3146 . . . . 5 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → ∀𝑗 ∈ (1...𝐾) ¬ (𝑌𝑗) = (𝑋𝐼))
213 ralnex 3072 . . . . . . . 8 (∀𝑗 ∈ (1...𝐾) ¬ (𝑌𝑗) = (𝑋𝐼) ↔ ¬ ∃𝑗 ∈ (1...𝐾)(𝑌𝑗) = (𝑋𝐼))
214213a1i 11 . . . . . . 7 (𝜑 → (∀𝑗 ∈ (1...𝐾) ¬ (𝑌𝑗) = (𝑋𝐼) ↔ ¬ ∃𝑗 ∈ (1...𝐾)(𝑌𝑗) = (𝑋𝐼)))
215 nnel 3056 . . . . . . . . . 10 (¬ (𝑋𝐼) ∉ ran 𝑌 ↔ (𝑋𝐼) ∈ ran 𝑌)
216215a1i 11 . . . . . . . . 9 (𝜑 → (¬ (𝑋𝐼) ∉ ran 𝑌 ↔ (𝑋𝐼) ∈ ran 𝑌))
217 fvelrnb 6952 . . . . . . . . . 10 (𝑌 Fn (1...𝐾) → ((𝑋𝐼) ∈ ran 𝑌 ↔ ∃𝑗 ∈ (1...𝐾)(𝑌𝑗) = (𝑋𝐼)))
21846, 217syl 17 . . . . . . . . 9 (𝜑 → ((𝑋𝐼) ∈ ran 𝑌 ↔ ∃𝑗 ∈ (1...𝐾)(𝑌𝑗) = (𝑋𝐼)))
219216, 218bitrd 278 . . . . . . . 8 (𝜑 → (¬ (𝑋𝐼) ∉ ran 𝑌 ↔ ∃𝑗 ∈ (1...𝐾)(𝑌𝑗) = (𝑋𝐼)))
220219con1bid 355 . . . . . . 7 (𝜑 → (¬ ∃𝑗 ∈ (1...𝐾)(𝑌𝑗) = (𝑋𝐼) ↔ (𝑋𝐼) ∉ ran 𝑌))
221214, 220bitrd 278 . . . . . 6 (𝜑 → (∀𝑗 ∈ (1...𝐾) ¬ (𝑌𝑗) = (𝑋𝐼) ↔ (𝑋𝐼) ∉ ran 𝑌))
222221adantr 481 . . . . 5 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → (∀𝑗 ∈ (1...𝐾) ¬ (𝑌𝑗) = (𝑋𝐼) ↔ (𝑋𝐼) ∉ ran 𝑌))
223212, 222mpbid 231 . . . 4 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → (𝑋𝐼) ∉ ran 𝑌)
224 elnelne1 3057 . . . 4 (((𝑋𝐼) ∈ ran 𝑋 ∧ (𝑋𝐼) ∉ ran 𝑌) → ran 𝑋 ≠ ran 𝑌)
22599, 223, 224syl2anc 584 . . 3 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → ran 𝑋 ≠ ran 𝑌)
22645ffund 6721 . . . . . 6 (𝜑 → Fun 𝑌)
227226adantr 481 . . . . 5 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → Fun 𝑌)
22845fdmd 6728 . . . . . . 7 (𝜑 → dom 𝑌 = (1...𝐾))
22970, 228eleqtrrd 2836 . . . . . 6 (𝜑𝐼 ∈ dom 𝑌)
230229adantr 481 . . . . 5 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → 𝐼 ∈ dom 𝑌)
231 fvelrn 7078 . . . . 5 ((Fun 𝑌𝐼 ∈ dom 𝑌) → (𝑌𝐼) ∈ ran 𝑌)
232227, 230, 231syl2anc 584 . . . 4 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → (𝑌𝐼) ∈ ran 𝑌)
2331003ad2ant3 1135 . . . . . . . . . . 11 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝑗 ∈ ℕ)
234233nnred 12229 . . . . . . . . . 10 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝑗 ∈ ℝ)
2351033ad2ant1 1133 . . . . . . . . . 10 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝐼 ∈ ℝ)
236234, 235lttri4d 11357 . . . . . . . . 9 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑗 < 𝐼𝑗 = 𝐼𝐼 < 𝑗))
237313ad2ant1 1133 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝑋:(1...𝐾)⟶(1...𝑁))
238 simp3 1138 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝑗 ∈ (1...𝐾))
239237, 238ffvelcdmd 7087 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝑗) ∈ (1...𝑁))
240109sseli 3978 . . . . . . . . . . . . . 14 ((𝑋𝑗) ∈ (1...𝑁) → (𝑋𝑗) ∈ ℕ)
241239, 240syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝑗) ∈ ℕ)
242241nnred 12229 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝑗) ∈ ℝ)
243242adantr 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑋𝑗) ∈ ℝ)
2441883ad2ant1 1133 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)))
245244adantr 481 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)))
246 simpl3 1193 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → 𝑗 ∈ (1...𝐾))
247703ad2ant1 1133 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝐼 ∈ (1...𝐾))
248247adantr 481 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → 𝐼 ∈ (1...𝐾))
249 fveq2 6891 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑗 → (𝑌𝑥) = (𝑌𝑗))
250249breq1d 5158 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑗 → ((𝑌𝑥) < (𝑌𝑦) ↔ (𝑌𝑗) < (𝑌𝑦)))
251121, 250imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑗 → ((𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)) ↔ (𝑗 < 𝑦 → (𝑌𝑗) < (𝑌𝑦))))
252 fveq2 6891 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝐼 → (𝑌𝑦) = (𝑌𝐼))
253252breq2d 5160 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐼 → ((𝑌𝑗) < (𝑌𝑦) ↔ (𝑌𝑗) < (𝑌𝐼)))
254125, 253imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐼 → ((𝑗 < 𝑦 → (𝑌𝑗) < (𝑌𝑦)) ↔ (𝑗 < 𝐼 → (𝑌𝑗) < (𝑌𝐼))))
255251, 254rspc2v 3622 . . . . . . . . . . . . . . 15 ((𝑗 ∈ (1...𝐾) ∧ 𝐼 ∈ (1...𝐾)) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)) → (𝑗 < 𝐼 → (𝑌𝑗) < (𝑌𝐼))))
256246, 248, 255syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)) → (𝑗 < 𝐼 → (𝑌𝑗) < (𝑌𝐼))))
257245, 256mpd 15 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑗 < 𝐼 → (𝑌𝑗) < (𝑌𝐼)))
258257syldbl2 839 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑌𝑗) < (𝑌𝐼))
2591693adantl2 1167 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑋𝑗) = (𝑌𝑗))
260259breq1d 5158 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → ((𝑋𝑗) < (𝑌𝐼) ↔ (𝑌𝑗) < (𝑌𝐼)))
261258, 260mpbird 256 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑋𝑗) < (𝑌𝐼))
262243, 261ltned 11352 . . . . . . . . . 10 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑋𝑗) ≠ (𝑌𝐼))
263893ad2ant1 1133 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑌𝐼) ∈ ℝ)
264263adantr 481 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑌𝐼) ∈ ℝ)
265 simpl2 1192 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑌𝐼) < (𝑋𝐼))
266264, 265ltned 11352 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑌𝐼) ≠ (𝑋𝐼))
267266necomd 2996 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑋𝐼) ≠ (𝑌𝐼))
268 fveq2 6891 . . . . . . . . . . . . 13 (𝑗 = 𝐼 → (𝑋𝑗) = (𝑋𝐼))
269268neeq1d 3000 . . . . . . . . . . . 12 (𝑗 = 𝐼 → ((𝑋𝑗) ≠ (𝑌𝐼) ↔ (𝑋𝐼) ≠ (𝑌𝐼)))
270269adantl 482 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → ((𝑋𝑗) ≠ (𝑌𝐼) ↔ (𝑋𝐼) ≠ (𝑌𝐼)))
271267, 270mpbird 256 . . . . . . . . . 10 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑋𝑗) ≠ (𝑌𝐼))
272263adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝐼) ∈ ℝ)
273873ad2ant1 1133 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝐼) ∈ ℝ)
274273adantr 481 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝐼) ∈ ℝ)
275242adantr 481 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝑗) ∈ ℝ)
276 simpl2 1192 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝐼) < (𝑋𝐼))
2771153ad2ant1 1133 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)))
278277adantr 481 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)))
279247adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → 𝐼 ∈ (1...𝐾))
280238adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → 𝑗 ∈ (1...𝐾))
281 fveq2 6891 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝐼 → (𝑋𝑥) = (𝑋𝐼))
282281breq1d 5158 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐼 → ((𝑋𝑥) < (𝑋𝑦) ↔ (𝑋𝐼) < (𝑋𝑦)))
283193, 282imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐼 → ((𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)) ↔ (𝐼 < 𝑦 → (𝑋𝐼) < (𝑋𝑦))))
284 fveq2 6891 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑗 → (𝑋𝑦) = (𝑋𝑗))
285284breq2d 5160 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑗 → ((𝑋𝐼) < (𝑋𝑦) ↔ (𝑋𝐼) < (𝑋𝑗)))
286197, 285imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑗 → ((𝐼 < 𝑦 → (𝑋𝐼) < (𝑋𝑦)) ↔ (𝐼 < 𝑗 → (𝑋𝐼) < (𝑋𝑗))))
287283, 286rspc2v 3622 . . . . . . . . . . . . . . . 16 ((𝐼 ∈ (1...𝐾) ∧ 𝑗 ∈ (1...𝐾)) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)) → (𝐼 < 𝑗 → (𝑋𝐼) < (𝑋𝑗))))
288279, 280, 287syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)) → (𝐼 < 𝑗 → (𝑋𝐼) < (𝑋𝑗))))
289278, 288mpd 15 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝐼 < 𝑗 → (𝑋𝐼) < (𝑋𝑗)))
290289syldbl2 839 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝐼) < (𝑋𝑗))
291272, 274, 275, 276, 290lttrd 11377 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝐼) < (𝑋𝑗))
292272, 291ltned 11352 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝐼) ≠ (𝑋𝑗))
293292necomd 2996 . . . . . . . . . 10 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝑗) ≠ (𝑌𝐼))
294262, 271, 2933jaodan 1430 . . . . . . . . 9 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ (𝑗 < 𝐼𝑗 = 𝐼𝐼 < 𝑗)) → (𝑋𝑗) ≠ (𝑌𝐼))
295236, 294mpdan 685 . . . . . . . 8 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝑗) ≠ (𝑌𝐼))
2962953expa 1118 . . . . . . 7 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝑗) ≠ (𝑌𝐼))
297296neneqd 2945 . . . . . 6 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) ∧ 𝑗 ∈ (1...𝐾)) → ¬ (𝑋𝑗) = (𝑌𝐼))
298297ralrimiva 3146 . . . . 5 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → ∀𝑗 ∈ (1...𝐾) ¬ (𝑋𝑗) = (𝑌𝐼))
299 ralnex 3072 . . . . . . . 8 (∀𝑗 ∈ (1...𝐾) ¬ (𝑋𝑗) = (𝑌𝐼) ↔ ¬ ∃𝑗 ∈ (1...𝐾)(𝑋𝑗) = (𝑌𝐼))
300299a1i 11 . . . . . . 7 (𝜑 → (∀𝑗 ∈ (1...𝐾) ¬ (𝑋𝑗) = (𝑌𝐼) ↔ ¬ ∃𝑗 ∈ (1...𝐾)(𝑋𝑗) = (𝑌𝐼)))
301 nnel 3056 . . . . . . . . . 10 (¬ (𝑌𝐼) ∉ ran 𝑋 ↔ (𝑌𝐼) ∈ ran 𝑋)
302301a1i 11 . . . . . . . . 9 (𝜑 → (¬ (𝑌𝐼) ∉ ran 𝑋 ↔ (𝑌𝐼) ∈ ran 𝑋))
303 fvelrnb 6952 . . . . . . . . . 10 (𝑋 Fn (1...𝐾) → ((𝑌𝐼) ∈ ran 𝑋 ↔ ∃𝑗 ∈ (1...𝐾)(𝑋𝑗) = (𝑌𝐼)))
30432, 303syl 17 . . . . . . . . 9 (𝜑 → ((𝑌𝐼) ∈ ran 𝑋 ↔ ∃𝑗 ∈ (1...𝐾)(𝑋𝑗) = (𝑌𝐼)))
305302, 304bitrd 278 . . . . . . . 8 (𝜑 → (¬ (𝑌𝐼) ∉ ran 𝑋 ↔ ∃𝑗 ∈ (1...𝐾)(𝑋𝑗) = (𝑌𝐼)))
306305con1bid 355 . . . . . . 7 (𝜑 → (¬ ∃𝑗 ∈ (1...𝐾)(𝑋𝑗) = (𝑌𝐼) ↔ (𝑌𝐼) ∉ ran 𝑋))
307300, 306bitrd 278 . . . . . 6 (𝜑 → (∀𝑗 ∈ (1...𝐾) ¬ (𝑋𝑗) = (𝑌𝐼) ↔ (𝑌𝐼) ∉ ran 𝑋))
308307adantr 481 . . . . 5 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → (∀𝑗 ∈ (1...𝐾) ¬ (𝑋𝑗) = (𝑌𝐼) ↔ (𝑌𝐼) ∉ ran 𝑋))
309298, 308mpbid 231 . . . 4 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → (𝑌𝐼) ∉ ran 𝑋)
310 elnelne1 3057 . . . . 5 (((𝑌𝐼) ∈ ran 𝑌 ∧ (𝑌𝐼) ∉ ran 𝑋) → ran 𝑌 ≠ ran 𝑋)
311310necomd 2996 . . . 4 (((𝑌𝐼) ∈ ran 𝑌 ∧ (𝑌𝐼) ∉ ran 𝑋) → ran 𝑋 ≠ ran 𝑌)
312232, 309, 311syl2anc 584 . . 3 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → ran 𝑋 ≠ ran 𝑌)
313225, 312jaodan 956 . 2 ((𝜑 ∧ ((𝑋𝐼) < (𝑌𝐼) ∨ (𝑌𝐼) < (𝑋𝐼))) → ran 𝑋 ≠ ran 𝑌)
31492, 313mpdan 685 1 (𝜑 → ran 𝑋 ≠ ran 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845  w3o 1086  w3a 1087  wal 1539   = wceq 1541  wcel 2106  {cab 2709  wne 2940  wnel 3046  wral 3061  wrex 3070  {crab 3432  wss 3948  c0 4322   class class class wbr 5148   Or wor 5587  dom cdm 5676  ran crn 5677  Fun wfun 6537   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7411  Fincfn 8941  infcinf 9438  cr 11111  1c1 11113   < clt 11250  cle 11251  cn 12214  0cn0 12474  ...cfz 13486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-n0 12475  df-z 12561  df-uz 12825  df-fz 13487
This theorem is referenced by:  sticksstones2  41055
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