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Theorem sticksstones1 40554
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 11235 . . . . . . 7 < Or ℝ
43a1i 11 . . . . . 6 (𝜑 → < Or ℝ)
5 fzfid 13878 . . . . . . . 8 (𝜑 → (1...𝐾) ∈ Fin)
6 ssrab2 4037 . . . . . . . . 9 {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ⊆ (1...𝐾)
76a1i 11 . . . . . . . 8 (𝜑 → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ⊆ (1...𝐾))
8 ssfi 9117 . . . . . . . 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 4344 . . . . . . . . . . . . 13 ({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} = ∅ ↔ ∀𝑧 ∈ (1...𝐾) ¬ (𝑋𝑧) ≠ (𝑌𝑧))
12 nne 2947 . . . . . . . . . . . . . 14 (¬ (𝑋𝑧) ≠ (𝑌𝑧) ↔ (𝑋𝑧) = (𝑌𝑧))
1312ralbii 3096 . . . . . . . . . . . . 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 6649 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑋 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑋:(1...𝐾)⟶(1...𝑁)))
16 fveq1 6841 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 = 𝑋 → (𝑓𝑥) = (𝑋𝑥))
17 fveq1 6841 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 = 𝑋 → (𝑓𝑦) = (𝑋𝑦))
1816, 17breq12d 5118 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑋 → ((𝑓𝑥) < (𝑓𝑦) ↔ (𝑋𝑥) < (𝑋𝑦)))
1918imbi2d 340 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = 𝑋 → ((𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ (𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦))))
20192ralbidv 3212 . . . . . . . . . . . . . . . . . . . . 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 eqab 2877 . . . . . . . . . . . . . . . . . . . . . . . . 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 3143 . . . . . . . . . . . . . . . . . . . 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 3582 . . . . . . . . . . . . . . . . . . 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 6669 . . . . . . . . . . . . . . . . 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 6649 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑌 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑌:(1...𝐾)⟶(1...𝑁)))
36 fveq1 6841 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = 𝑌 → (𝑓𝑥) = (𝑌𝑥))
37 fveq1 6841 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = 𝑌 → (𝑓𝑦) = (𝑌𝑦))
3836, 37breq12d 5118 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 = 𝑌 → ((𝑓𝑥) < (𝑓𝑦) ↔ (𝑌𝑥) < (𝑌𝑦)))
3938imbi2d 340 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 = 𝑌 → ((𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ (𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦))))
40392ralbidv 3212 . . . . . . . . . . . . . . . . . . . . . . 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 3582 . . . . . . . . . . . . . . . . . . . . 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 6669 . . . . . . . . . . . . . . . . 17 (𝜑𝑌 Fn (1...𝐾))
4746adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧)) → 𝑌 Fn (1...𝐾))
48 eqfnfv 6982 . . . . . . . . . . . . . . . 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 2964 . . . . . . . . 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 13472 . . . . . . . . . 10 (1...𝐾) ⊆ ℕ
5958a1i 11 . . . . . . . . 9 (𝜑 → (1...𝐾) ⊆ ℕ)
60 nnssre 12157 . . . . . . . . . 10 ℕ ⊆ ℝ
6160a1i 11 . . . . . . . . 9 (𝜑 → ℕ ⊆ ℝ)
6259, 61sstrd 3954 . . . . . . . 8 (𝜑 → (1...𝐾) ⊆ ℝ)
637, 62sstrd 3954 . . . . . . 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 9437 . . . . . 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 2838 . . . 4 (𝜑𝐼 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)})
687, 66sseldd 3945 . . . . . 6 (𝜑 → inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < ) ∈ (1...𝐾))
692eleq1d 2822 . . . . . 6 (𝜑 → (𝐼 ∈ (1...𝐾) ↔ inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < ) ∈ (1...𝐾)))
7068, 69mpbird 256 . . . . 5 (𝜑𝐼 ∈ (1...𝐾))
71 fveq2 6842 . . . . . . 7 (𝑧 = 𝐼 → (𝑋𝑧) = (𝑋𝐼))
72 fveq2 6842 . . . . . . 7 (𝑧 = 𝐼 → (𝑌𝑧) = (𝑌𝐼))
7371, 72neeq12d 3005 . . . . . 6 (𝑧 = 𝐼 → ((𝑋𝑧) ≠ (𝑌𝑧) ↔ (𝑋𝐼) ≠ (𝑌𝐼)))
7473elrab3 3646 . . . . 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 2907 . . . . . 6 𝑎(1...𝑁)
79 nfcv 2907 . . . . . 6 𝑎
80 elfznn 13470 . . . . . . . . 9 (𝑎 ∈ (1...𝑁) → 𝑎 ∈ ℕ)
8180adantl 482 . . . . . . . 8 ((𝜑𝑎 ∈ (1...𝑁)) → 𝑎 ∈ ℕ)
82 nnre 12160 . . . . . . . 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 3949 . . . . 5 (𝜑 → (1...𝑁) ⊆ ℝ)
8631, 70ffvelcdmd 7036 . . . . 5 (𝜑 → (𝑋𝐼) ∈ (1...𝑁))
8785, 86sseldd 3945 . . . 4 (𝜑 → (𝑋𝐼) ∈ ℝ)
8845, 70ffvelcdmd 7036 . . . . 5 (𝜑 → (𝑌𝐼) ∈ (1...𝑁))
8985, 88sseldd 3945 . . . 4 (𝜑 → (𝑌𝐼) ∈ ℝ)
90 lttri2 11237 . . . 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 6672 . . . . . 6 (𝜑 → Fun 𝑋)
9493adantr 481 . . . . 5 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → Fun 𝑋)
9531fdmd 6679 . . . . . . 7 (𝜑 → dom 𝑋 = (1...𝐾))
9670, 95eleqtrrd 2841 . . . . . 6 (𝜑𝐼 ∈ dom 𝑋)
9796adantr 481 . . . . 5 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → 𝐼 ∈ dom 𝑋)
98 fvelrn 7027 . . . . 5 ((Fun 𝑋𝐼 ∈ dom 𝑋) → (𝑋𝐼) ∈ ran 𝑋)
9994, 97, 98syl2anc 584 . . . 4 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → (𝑋𝐼) ∈ ran 𝑋)
100 elfznn 13470 . . . . . . . . . . . 12 (𝑗 ∈ (1...𝐾) → 𝑗 ∈ ℕ)
1011003ad2ant3 1135 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝑗 ∈ ℕ)
102101nnred 12168 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝑗 ∈ ℝ)
10362, 70sseldd 3945 . . . . . . . . . . 11 (𝜑𝐼 ∈ ℝ)
1041033ad2ant1 1133 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝐼 ∈ ℝ)
105102, 104lttri4d 11296 . . . . . . . . 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 7036 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑌𝑗) ∈ (1...𝑁))
109 fz1ssnn 13472 . . . . . . . . . . . . . . 15 (1...𝑁) ⊆ ℕ
110109sseli 3940 . . . . . . . . . . . . . 14 ((𝑌𝑗) ∈ (1...𝑁) → (𝑌𝑗) ∈ ℕ)
111 nnre 12160 . . . . . . . . . . . . . 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 5108 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑗 → (𝑥 < 𝑦𝑗 < 𝑦))
122 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑗 → (𝑋𝑥) = (𝑋𝑗))
123122breq1d 5115 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑗 → ((𝑋𝑥) < (𝑋𝑦) ↔ (𝑋𝑗) < (𝑋𝑦)))
124121, 123imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑗 → ((𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)) ↔ (𝑗 < 𝑦 → (𝑋𝑗) < (𝑋𝑦))))
125 breq2 5109 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐼 → (𝑗 < 𝑦𝑗 < 𝐼))
126 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝐼 → (𝑋𝑦) = (𝑋𝐼))
127126breq2d 5117 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐼 → ((𝑋𝑗) < (𝑋𝑦) ↔ (𝑋𝑗) < (𝑋𝐼)))
128125, 127imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐼 → ((𝑗 < 𝑦 → (𝑋𝑗) < (𝑋𝑦)) ↔ (𝑗 < 𝐼 → (𝑋𝑗) < (𝑋𝐼))))
129124, 128rspc2v 3590 . . . . . . . . . . . . . . 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 12168 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → 𝑗 ∈ ℝ)
1371033ad2ant1 1133 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → 𝐼 ∈ ℝ)
138136, 137ltnled 11302 . . . . . . . . . . . . . . . . . . . . . . 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 12141 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5115 . . . . . . . . . . . . . . . . . . . . . . . . 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 2907 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑧𝑗
153 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑧(1...𝐾)
154 nfv 1917 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑧(𝑋𝑗) ≠ (𝑌𝑗)
155 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = 𝑗 → (𝑋𝑧) = (𝑋𝑗))
156 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = 𝑗 → (𝑌𝑧) = (𝑌𝑗))
157155, 156neeq12d 3005 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 = 𝑗 → ((𝑋𝑧) ≠ (𝑌𝑧) ↔ (𝑋𝑗) ≠ (𝑌𝑗)))
158152, 153, 154, 157elrabf 3641 . . . . . . . . . . . . . . . . . . . . . . 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 2947 . . . . . . . . . . . . . . . . . 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 2742 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑌𝑗) = (𝑋𝑗))
172171breq1d 5115 . . . . . . . . . . . 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 11291 . . . . . . . . . 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 2999 . . . . . . . . . . 11 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑌𝐼) ≠ (𝑋𝐼))
178 fveq2 6842 . . . . . . . . . . . . 13 (𝑗 = 𝐼 → (𝑌𝑗) = (𝑌𝐼))
179178neeq1d 3003 . . . . . . . . . . . 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 5108 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐼 → (𝑥 < 𝑦𝐼 < 𝑦))
194 fveq2 6842 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝐼 → (𝑌𝑥) = (𝑌𝐼))
195194breq1d 5115 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐼 → ((𝑌𝑥) < (𝑌𝑦) ↔ (𝑌𝐼) < (𝑌𝑦)))
196193, 195imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐼 → ((𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)) ↔ (𝐼 < 𝑦 → (𝑌𝐼) < (𝑌𝑦))))
197 breq2 5109 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑗 → (𝐼 < 𝑦𝐼 < 𝑗))
198 fveq2 6842 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑗 → (𝑌𝑦) = (𝑌𝑗))
199198breq2d 5117 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑗 → ((𝑌𝐼) < (𝑌𝑦) ↔ (𝑌𝐼) < (𝑌𝑗)))
200197, 199imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑗 → ((𝐼 < 𝑦 → (𝑌𝐼) < (𝑌𝑦)) ↔ (𝐼 < 𝑗 → (𝑌𝐼) < (𝑌𝑗))))
201196, 200rspc2v 3590 . . . . . . . . . . . . . . . 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 11316 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝐼) < (𝑌𝑗))
206183, 205ltned 11291 . . . . . . . . . . 11 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝐼) ≠ (𝑌𝑗))
207206necomd 2999 . . . . . . . . . 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 2948 . . . . . 6 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) ∧ 𝑗 ∈ (1...𝐾)) → ¬ (𝑌𝑗) = (𝑋𝐼))
212211ralrimiva 3143 . . . . 5 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → ∀𝑗 ∈ (1...𝐾) ¬ (𝑌𝑗) = (𝑋𝐼))
213 ralnex 3075 . . . . . . . 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 3058 . . . . . . . . . 10 (¬ (𝑋𝐼) ∉ ran 𝑌 ↔ (𝑋𝐼) ∈ ran 𝑌)
216215a1i 11 . . . . . . . . 9 (𝜑 → (¬ (𝑋𝐼) ∉ ran 𝑌 ↔ (𝑋𝐼) ∈ ran 𝑌))
217 fvelrnb 6903 . . . . . . . . . 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 3059 . . . 4 (((𝑋𝐼) ∈ ran 𝑋 ∧ (𝑋𝐼) ∉ ran 𝑌) → ran 𝑋 ≠ ran 𝑌)
22599, 223, 224syl2anc 584 . . 3 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → ran 𝑋 ≠ ran 𝑌)
22645ffund 6672 . . . . . 6 (𝜑 → Fun 𝑌)
227226adantr 481 . . . . 5 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → Fun 𝑌)
22845fdmd 6679 . . . . . . 7 (𝜑 → dom 𝑌 = (1...𝐾))
22970, 228eleqtrrd 2841 . . . . . 6 (𝜑𝐼 ∈ dom 𝑌)
230229adantr 481 . . . . 5 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → 𝐼 ∈ dom 𝑌)
231 fvelrn 7027 . . . . 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 12168 . . . . . . . . . 10 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝑗 ∈ ℝ)
2351033ad2ant1 1133 . . . . . . . . . 10 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝐼 ∈ ℝ)
236234, 235lttri4d 11296 . . . . . . . . 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 7036 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝑗) ∈ (1...𝑁))
240109sseli 3940 . . . . . . . . . . . . . 14 ((𝑋𝑗) ∈ (1...𝑁) → (𝑋𝑗) ∈ ℕ)
241239, 240syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝑗) ∈ ℕ)
242241nnred 12168 . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑗 → (𝑌𝑥) = (𝑌𝑗))
250249breq1d 5115 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑗 → ((𝑌𝑥) < (𝑌𝑦) ↔ (𝑌𝑗) < (𝑌𝑦)))
251121, 250imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑗 → ((𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)) ↔ (𝑗 < 𝑦 → (𝑌𝑗) < (𝑌𝑦))))
252 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝐼 → (𝑌𝑦) = (𝑌𝐼))
253252breq2d 5117 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐼 → ((𝑌𝑗) < (𝑌𝑦) ↔ (𝑌𝑗) < (𝑌𝐼)))
254125, 253imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐼 → ((𝑗 < 𝑦 → (𝑌𝑗) < (𝑌𝑦)) ↔ (𝑗 < 𝐼 → (𝑌𝑗) < (𝑌𝐼))))
255251, 254rspc2v 3590 . . . . . . . . . . . . . . 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 5115 . . . . . . . . . . . 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 11291 . . . . . . . . . 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 11291 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑌𝐼) ≠ (𝑋𝐼))
267266necomd 2999 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑋𝐼) ≠ (𝑌𝐼))
268 fveq2 6842 . . . . . . . . . . . . 13 (𝑗 = 𝐼 → (𝑋𝑗) = (𝑋𝐼))
269268neeq1d 3003 . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝐼 → (𝑋𝑥) = (𝑋𝐼))
282281breq1d 5115 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐼 → ((𝑋𝑥) < (𝑋𝑦) ↔ (𝑋𝐼) < (𝑋𝑦)))
283193, 282imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐼 → ((𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)) ↔ (𝐼 < 𝑦 → (𝑋𝐼) < (𝑋𝑦))))
284 fveq2 6842 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑗 → (𝑋𝑦) = (𝑋𝑗))
285284breq2d 5117 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑗 → ((𝑋𝐼) < (𝑋𝑦) ↔ (𝑋𝐼) < (𝑋𝑗)))
286197, 285imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑗 → ((𝐼 < 𝑦 → (𝑋𝐼) < (𝑋𝑦)) ↔ (𝐼 < 𝑗 → (𝑋𝐼) < (𝑋𝑗))))
287283, 286rspc2v 3590 . . . . . . . . . . . . . . . 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 11316 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝐼) < (𝑋𝑗))
292272, 291ltned 11291 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝐼) ≠ (𝑋𝑗))
293292necomd 2999 . . . . . . . . . 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 2948 . . . . . 6 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) ∧ 𝑗 ∈ (1...𝐾)) → ¬ (𝑋𝑗) = (𝑌𝐼))
298297ralrimiva 3143 . . . . 5 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → ∀𝑗 ∈ (1...𝐾) ¬ (𝑋𝑗) = (𝑌𝐼))
299 ralnex 3075 . . . . . . . 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 3058 . . . . . . . . . 10 (¬ (𝑌𝐼) ∉ ran 𝑋 ↔ (𝑌𝐼) ∈ ran 𝑋)
302301a1i 11 . . . . . . . . 9 (𝜑 → (¬ (𝑌𝐼) ∉ ran 𝑋 ↔ (𝑌𝐼) ∈ ran 𝑋))
303 fvelrnb 6903 . . . . . . . . . 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 3059 . . . . 5 (((𝑌𝐼) ∈ ran 𝑌 ∧ (𝑌𝐼) ∉ ran 𝑋) → ran 𝑌 ≠ ran 𝑋)
311310necomd 2999 . . . 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 2713  wne 2943  wnel 3049  wral 3064  wrex 3073  {crab 3407  wss 3910  c0 4282   class class class wbr 5105   Or wor 5544  dom cdm 5633  ran crn 5634  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  Fincfn 8883  infcinf 9377  cr 11050  1c1 11052   < clt 11189  cle 11190  cn 12153  0cn0 12413  ...cfz 13424
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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425
This theorem is referenced by:  sticksstones2  40555
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