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Theorem sticksstones1 40962
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 11294 . . . . . . 7 < Or ℝ
43a1i 11 . . . . . 6 (𝜑 → < Or ℝ)
5 fzfid 13938 . . . . . . . 8 (𝜑 → (1...𝐾) ∈ Fin)
6 ssrab2 4078 . . . . . . . . 9 {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ⊆ (1...𝐾)
76a1i 11 . . . . . . . 8 (𝜑 → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ⊆ (1...𝐾))
8 ssfi 9173 . . . . . . . 8 (((1...𝐾) ∈ Fin ∧ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ⊆ (1...𝐾)) → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ∈ Fin)
95, 7, 8syl2anc 585 . . . . . . 7 (𝜑 → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ∈ Fin)
10 sticksstones1.6 . . . . . . . 8 (𝜑𝑋𝑌)
11 rabeq0 4385 . . . . . . . . . . . . 13 ({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} = ∅ ↔ ∀𝑧 ∈ (1...𝐾) ¬ (𝑋𝑧) ≠ (𝑌𝑧))
12 nne 2945 . . . . . . . . . . . . . 14 (¬ (𝑋𝑧) ≠ (𝑌𝑧) ↔ (𝑋𝑧) = (𝑌𝑧))
1312ralbii 3094 . . . . . . . . . . . . 13 (∀𝑧 ∈ (1...𝐾) ¬ (𝑋𝑧) ≠ (𝑌𝑧) ↔ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧))
1411, 13bitri 275 . . . . . . . . . . . 12 ({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} = ∅ ↔ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧))
15 feq1 6699 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑋 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑋:(1...𝐾)⟶(1...𝑁)))
16 fveq1 6891 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 = 𝑋 → (𝑓𝑥) = (𝑋𝑥))
17 fveq1 6891 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 = 𝑋 → (𝑓𝑦) = (𝑋𝑦))
1816, 17breq12d 5162 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑋 → ((𝑓𝑥) < (𝑓𝑦) ↔ (𝑋𝑥) < (𝑋𝑦)))
1918imbi2d 341 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = 𝑋 → ((𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ (𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦))))
20192ralbidv 3219 . . . . . . . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . . . . . . . . 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 2874 . . . . . . . . . . . . . . . . . . . . . . . . 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 2178 . . . . . . . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑓𝐴) → (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦))))
2827ralrimiva 3147 . . . . . . . . . . . . . . . . . . . 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 3614 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑋:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦))))
3130simpld 496 . . . . . . . . . . . . . . . . . 18 (𝜑𝑋:(1...𝐾)⟶(1...𝑁))
3231ffnd 6719 . . . . . . . . . . . . . . . . 17 (𝜑𝑋 Fn (1...𝐾))
3332adantr 482 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧)) → 𝑋 Fn (1...𝐾))
34 sticksstones1.5 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑌𝐴)
35 feq1 6699 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑌 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑌:(1...𝐾)⟶(1...𝑁)))
36 fveq1 6891 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = 𝑌 → (𝑓𝑥) = (𝑌𝑥))
37 fveq1 6891 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = 𝑌 → (𝑓𝑦) = (𝑌𝑦))
3836, 37breq12d 5162 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 = 𝑌 → ((𝑓𝑥) < (𝑓𝑦) ↔ (𝑌𝑥) < (𝑌𝑦)))
3938imbi2d 341 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 = 𝑌 → ((𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ (𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦))))
40392ralbidv 3219 . . . . . . . . . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . . . . . . . . . . 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 3614 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑌:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦))))
4342adantr 482 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑌𝐴) → (𝑌:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦))))
4434, 43mpdan 686 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑌:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦))))
4544simpld 496 . . . . . . . . . . . . . . . . . 18 (𝜑𝑌:(1...𝐾)⟶(1...𝑁))
4645ffnd 6719 . . . . . . . . . . . . . . . . 17 (𝜑𝑌 Fn (1...𝐾))
4746adantr 482 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧)) → 𝑌 Fn (1...𝐾))
48 eqfnfv 7033 . . . . . . . . . . . . . . . 16 ((𝑋 Fn (1...𝐾) ∧ 𝑌 Fn (1...𝐾)) → (𝑋 = 𝑌 ↔ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧)))
4933, 47, 48syl2anc 585 . . . . . . . . . . . . . . 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 840 . . . . . . . . . . . 12 ((𝜑 ∧ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧)) → 𝑋 = 𝑌)
5314, 52sylan2b 595 . . . . . . . . . . 11 ((𝜑 ∧ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} = ∅) → 𝑋 = 𝑌)
5453ex 414 . . . . . . . . . 10 (𝜑 → ({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} = ∅ → 𝑋 = 𝑌))
5554necon3d 2962 . . . . . . . . 9 (𝜑 → (𝑋𝑌 → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ≠ ∅))
5655imp 408 . . . . . . . 8 ((𝜑𝑋𝑌) → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ≠ ∅)
5710, 56mpdan 686 . . . . . . 7 (𝜑 → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ≠ ∅)
58 fz1ssnn 13532 . . . . . . . . . 10 (1...𝐾) ⊆ ℕ
5958a1i 11 . . . . . . . . 9 (𝜑 → (1...𝐾) ⊆ ℕ)
60 nnssre 12216 . . . . . . . . . 10 ℕ ⊆ ℝ
6160a1i 11 . . . . . . . . 9 (𝜑 → ℕ ⊆ ℝ)
6259, 61sstrd 3993 . . . . . . . 8 (𝜑 → (1...𝐾) ⊆ ℝ)
637, 62sstrd 3993 . . . . . . 7 (𝜑 → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ⊆ ℝ)
649, 57, 633jca 1129 . . . . . 6 (𝜑 → ({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ∈ Fin ∧ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ≠ ∅ ∧ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ⊆ ℝ))
65 fiinfcl 9496 . . . . . 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 585 . . . . 5 (𝜑 → inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < ) ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)})
672, 66eqeltrd 2834 . . . 4 (𝜑𝐼 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)})
687, 66sseldd 3984 . . . . . 6 (𝜑 → inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < ) ∈ (1...𝐾))
692eleq1d 2819 . . . . . 6 (𝜑 → (𝐼 ∈ (1...𝐾) ↔ inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < ) ∈ (1...𝐾)))
7068, 69mpbird 257 . . . . 5 (𝜑𝐼 ∈ (1...𝐾))
71 fveq2 6892 . . . . . . 7 (𝑧 = 𝐼 → (𝑋𝑧) = (𝑋𝐼))
72 fveq2 6892 . . . . . . 7 (𝑧 = 𝐼 → (𝑌𝑧) = (𝑌𝐼))
7371, 72neeq12d 3003 . . . . . 6 (𝑧 = 𝐼 → ((𝑋𝑧) ≠ (𝑌𝑧) ↔ (𝑋𝐼) ≠ (𝑌𝐼)))
7473elrab3 3685 . . . . 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 1918 . . . . . 6 𝑎𝜑
78 nfcv 2904 . . . . . 6 𝑎(1...𝑁)
79 nfcv 2904 . . . . . 6 𝑎
80 elfznn 13530 . . . . . . . . 9 (𝑎 ∈ (1...𝑁) → 𝑎 ∈ ℕ)
8180adantl 483 . . . . . . . 8 ((𝜑𝑎 ∈ (1...𝑁)) → 𝑎 ∈ ℕ)
82 nnre 12219 . . . . . . . 8 (𝑎 ∈ ℕ → 𝑎 ∈ ℝ)
8381, 82syl 17 . . . . . . 7 ((𝜑𝑎 ∈ (1...𝑁)) → 𝑎 ∈ ℝ)
8483ex 414 . . . . . 6 (𝜑 → (𝑎 ∈ (1...𝑁) → 𝑎 ∈ ℝ))
8577, 78, 79, 84ssrd 3988 . . . . 5 (𝜑 → (1...𝑁) ⊆ ℝ)
8631, 70ffvelcdmd 7088 . . . . 5 (𝜑 → (𝑋𝐼) ∈ (1...𝑁))
8785, 86sseldd 3984 . . . 4 (𝜑 → (𝑋𝐼) ∈ ℝ)
8845, 70ffvelcdmd 7088 . . . . 5 (𝜑 → (𝑌𝐼) ∈ (1...𝑁))
8985, 88sseldd 3984 . . . 4 (𝜑 → (𝑌𝐼) ∈ ℝ)
90 lttri2 11296 . . . 4 (((𝑋𝐼) ∈ ℝ ∧ (𝑌𝐼) ∈ ℝ) → ((𝑋𝐼) ≠ (𝑌𝐼) ↔ ((𝑋𝐼) < (𝑌𝐼) ∨ (𝑌𝐼) < (𝑋𝐼))))
9187, 89, 90syl2anc 585 . . 3 (𝜑 → ((𝑋𝐼) ≠ (𝑌𝐼) ↔ ((𝑋𝐼) < (𝑌𝐼) ∨ (𝑌𝐼) < (𝑋𝐼))))
9276, 91mpbid 231 . 2 (𝜑 → ((𝑋𝐼) < (𝑌𝐼) ∨ (𝑌𝐼) < (𝑋𝐼)))
9331ffund 6722 . . . . . 6 (𝜑 → Fun 𝑋)
9493adantr 482 . . . . 5 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → Fun 𝑋)
9531fdmd 6729 . . . . . . 7 (𝜑 → dom 𝑋 = (1...𝐾))
9670, 95eleqtrrd 2837 . . . . . 6 (𝜑𝐼 ∈ dom 𝑋)
9796adantr 482 . . . . 5 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → 𝐼 ∈ dom 𝑋)
98 fvelrn 7079 . . . . 5 ((Fun 𝑋𝐼 ∈ dom 𝑋) → (𝑋𝐼) ∈ ran 𝑋)
9994, 97, 98syl2anc 585 . . . 4 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → (𝑋𝐼) ∈ ran 𝑋)
100 elfznn 13530 . . . . . . . . . . . 12 (𝑗 ∈ (1...𝐾) → 𝑗 ∈ ℕ)
1011003ad2ant3 1136 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝑗 ∈ ℕ)
102101nnred 12227 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝑗 ∈ ℝ)
10362, 70sseldd 3984 . . . . . . . . . . 11 (𝜑𝐼 ∈ ℝ)
1041033ad2ant1 1134 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝐼 ∈ ℝ)
105102, 104lttri4d 11355 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑗 < 𝐼𝑗 = 𝐼𝐼 < 𝑗))
106453ad2ant1 1134 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝑌:(1...𝐾)⟶(1...𝑁))
107 simp3 1139 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝑗 ∈ (1...𝐾))
108106, 107ffvelcdmd 7088 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑌𝑗) ∈ (1...𝑁))
109 fz1ssnn 13532 . . . . . . . . . . . . . . 15 (1...𝑁) ⊆ ℕ
110109sseli 3979 . . . . . . . . . . . . . 14 ((𝑌𝑗) ∈ (1...𝑁) → (𝑌𝑗) ∈ ℕ)
111 nnre 12219 . . . . . . . . . . . . . 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 482 . . . . . . . . . . 11 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑌𝑗) ∈ ℝ)
11530simprd 497 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)))
1161153ad2ant1 1134 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)))
117116adantr 482 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)))
118 simpl3 1194 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → 𝑗 ∈ (1...𝐾))
119703ad2ant1 1134 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝐼 ∈ (1...𝐾))
120119adantr 482 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → 𝐼 ∈ (1...𝐾))
121 breq1 5152 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑗 → (𝑥 < 𝑦𝑗 < 𝑦))
122 fveq2 6892 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑗 → (𝑋𝑥) = (𝑋𝑗))
123122breq1d 5159 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑗 → ((𝑋𝑥) < (𝑋𝑦) ↔ (𝑋𝑗) < (𝑋𝑦)))
124121, 123imbi12d 345 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑗 → ((𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)) ↔ (𝑗 < 𝑦 → (𝑋𝑗) < (𝑋𝑦))))
125 breq2 5153 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐼 → (𝑗 < 𝑦𝑗 < 𝐼))
126 fveq2 6892 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝐼 → (𝑋𝑦) = (𝑋𝐼))
127126breq2d 5161 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐼 → ((𝑋𝑗) < (𝑋𝑦) ↔ (𝑋𝑗) < (𝑋𝐼)))
128125, 127imbi12d 345 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐼 → ((𝑗 < 𝑦 → (𝑋𝑗) < (𝑋𝑦)) ↔ (𝑗 < 𝐼 → (𝑋𝑗) < (𝑋𝐼))))
129124, 128rspc2v 3623 . . . . . . . . . . . . . . 15 ((𝑗 ∈ (1...𝐾) ∧ 𝐼 ∈ (1...𝐾)) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)) → (𝑗 < 𝐼 → (𝑋𝑗) < (𝑋𝐼))))
130118, 120, 129syl2anc 585 . . . . . . . . . . . . . 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 840 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑋𝑗) < (𝑋𝐼))
133 simp2 1138 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → 𝑗 ∈ (1...𝐾))
134 simp3 1139 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → 𝑗 < 𝐼)
1351003ad2ant2 1135 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → 𝑗 ∈ ℕ)
136135nnred 12227 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → 𝑗 ∈ ℝ)
1371033ad2ant1 1134 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → 𝐼 ∈ ℝ)
138136, 137ltnled 11361 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → (𝑗 < 𝐼 ↔ ¬ 𝐼𝑗))
139134, 138mpbid 231 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → ¬ 𝐼𝑗)
140633ad2ant1 1134 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ⊆ ℝ)
14193ad2ant1 1134 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ∈ Fin)
142 infrefilb 12200 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1122 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → (𝑗 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} → inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < ) ≤ 𝑗))
145144imp 408 . . . . . . . . . . . . . . . . . . . . . . . . 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 5159 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) ∧ 𝑗 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}) → (𝐼𝑗 ↔ inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < ) ≤ 𝑗))
148145, 147mpbird 257 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) ∧ 𝑗 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}) → 𝐼𝑗)
149148ex 414 . . . . . . . . . . . . . . . . . . . . . . 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 2904 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑧𝑗
153 nfcv 2904 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑧(1...𝐾)
154 nfv 1918 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑧(𝑋𝑗) ≠ (𝑌𝑗)
155 fveq2 6892 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = 𝑗 → (𝑋𝑧) = (𝑋𝑗))
156 fveq2 6892 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = 𝑗 → (𝑌𝑧) = (𝑌𝑗))
157155, 156neeq12d 3003 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 = 𝑗 → ((𝑋𝑧) ≠ (𝑌𝑧) ↔ (𝑋𝑗) ≠ (𝑌𝑗)))
158152, 153, 154, 157elrabf 3680 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ↔ (𝑗 ∈ (1...𝐾) ∧ (𝑋𝑗) ≠ (𝑌𝑗)))
159158notbii 320 . . . . . . . . . . . . . . . . . . . . . 22 𝑗 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ↔ ¬ (𝑗 ∈ (1...𝐾) ∧ (𝑋𝑗) ≠ (𝑌𝑗)))
160 ianor 981 . . . . . . . . . . . . . . . . . . . . . 22 (¬ (𝑗 ∈ (1...𝐾) ∧ (𝑋𝑗) ≠ (𝑌𝑗)) ↔ (¬ 𝑗 ∈ (1...𝐾) ∨ ¬ (𝑋𝑗) ≠ (𝑌𝑗)))
161159, 160bitri 275 . . . . . . . . . . . . . . . . . . . . 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 852 . . . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) ∧ 𝑗 ∈ (1...𝐾)) → ¬ (𝑋𝑗) ≠ (𝑌𝑗))
166 nne 2945 . . . . . . . . . . . . . . . . . 18 (¬ (𝑋𝑗) ≠ (𝑌𝑗) ↔ (𝑋𝑗) = (𝑌𝑗))
167165, 166sylib 217 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝑗) = (𝑌𝑗))
168133, 167mpdan 686 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → (𝑋𝑗) = (𝑌𝑗))
1691683expa 1119 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑋𝑗) = (𝑌𝑗))
1701693adantl2 1168 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑋𝑗) = (𝑌𝑗))
171170eqcomd 2739 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑌𝑗) = (𝑋𝑗))
172171breq1d 5159 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → ((𝑌𝑗) < (𝑋𝐼) ↔ (𝑋𝑗) < (𝑋𝐼)))
173132, 172mpbird 257 . . . . . . . . . . 11 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑌𝑗) < (𝑋𝐼))
174114, 173ltned 11350 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑌𝑗) ≠ (𝑋𝐼))
175763ad2ant1 1134 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝐼) ≠ (𝑌𝐼))
176175adantr 482 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑋𝐼) ≠ (𝑌𝐼))
177176necomd 2997 . . . . . . . . . . 11 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑌𝐼) ≠ (𝑋𝐼))
178 fveq2 6892 . . . . . . . . . . . . 13 (𝑗 = 𝐼 → (𝑌𝑗) = (𝑌𝐼))
179178neeq1d 3001 . . . . . . . . . . . 12 (𝑗 = 𝐼 → ((𝑌𝑗) ≠ (𝑋𝐼) ↔ (𝑌𝐼) ≠ (𝑋𝐼)))
180179adantl 483 . . . . . . . . . . 11 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → ((𝑌𝑗) ≠ (𝑋𝐼) ↔ (𝑌𝐼) ≠ (𝑋𝐼)))
181177, 180mpbird 257 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑌𝑗) ≠ (𝑋𝐼))
182873ad2ant1 1134 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝐼) ∈ ℝ)
183182adantr 482 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝐼) ∈ ℝ)
184893ad2ant1 1134 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑌𝐼) ∈ ℝ)
185184adantr 482 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝐼) ∈ ℝ)
186113adantr 482 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝑗) ∈ ℝ)
187 simpl2 1193 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝐼) < (𝑌𝐼))
18842simprd 497 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)))
1891883ad2ant1 1134 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)))
190189adantr 482 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)))
191119adantr 482 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → 𝐼 ∈ (1...𝐾))
192107adantr 482 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → 𝑗 ∈ (1...𝐾))
193 breq1 5152 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐼 → (𝑥 < 𝑦𝐼 < 𝑦))
194 fveq2 6892 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝐼 → (𝑌𝑥) = (𝑌𝐼))
195194breq1d 5159 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐼 → ((𝑌𝑥) < (𝑌𝑦) ↔ (𝑌𝐼) < (𝑌𝑦)))
196193, 195imbi12d 345 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐼 → ((𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)) ↔ (𝐼 < 𝑦 → (𝑌𝐼) < (𝑌𝑦))))
197 breq2 5153 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑗 → (𝐼 < 𝑦𝐼 < 𝑗))
198 fveq2 6892 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑗 → (𝑌𝑦) = (𝑌𝑗))
199198breq2d 5161 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑗 → ((𝑌𝐼) < (𝑌𝑦) ↔ (𝑌𝐼) < (𝑌𝑗)))
200197, 199imbi12d 345 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑗 → ((𝐼 < 𝑦 → (𝑌𝐼) < (𝑌𝑦)) ↔ (𝐼 < 𝑗 → (𝑌𝐼) < (𝑌𝑗))))
201196, 200rspc2v 3623 . . . . . . . . . . . . . . . 16 ((𝐼 ∈ (1...𝐾) ∧ 𝑗 ∈ (1...𝐾)) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)) → (𝐼 < 𝑗 → (𝑌𝐼) < (𝑌𝑗))))
202191, 192, 201syl2anc 585 . . . . . . . . . . . . . . 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 840 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝐼) < (𝑌𝑗))
205183, 185, 186, 187, 204lttrd 11375 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝐼) < (𝑌𝑗))
206183, 205ltned 11350 . . . . . . . . . . 11 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝐼) ≠ (𝑌𝑗))
207206necomd 2997 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝑗) ≠ (𝑋𝐼))
208174, 181, 2073jaodan 1431 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ (𝑗 < 𝐼𝑗 = 𝐼𝐼 < 𝑗)) → (𝑌𝑗) ≠ (𝑋𝐼))
209105, 208mpdan 686 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑌𝑗) ≠ (𝑋𝐼))
2102093expa 1119 . . . . . . 7 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) ∧ 𝑗 ∈ (1...𝐾)) → (𝑌𝑗) ≠ (𝑋𝐼))
211210neneqd 2946 . . . . . 6 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) ∧ 𝑗 ∈ (1...𝐾)) → ¬ (𝑌𝑗) = (𝑋𝐼))
212211ralrimiva 3147 . . . . 5 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → ∀𝑗 ∈ (1...𝐾) ¬ (𝑌𝑗) = (𝑋𝐼))
213 ralnex 3073 . . . . . . . 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 3057 . . . . . . . . . 10 (¬ (𝑋𝐼) ∉ ran 𝑌 ↔ (𝑋𝐼) ∈ ran 𝑌)
216215a1i 11 . . . . . . . . 9 (𝜑 → (¬ (𝑋𝐼) ∉ ran 𝑌 ↔ (𝑋𝐼) ∈ ran 𝑌))
217 fvelrnb 6953 . . . . . . . . . 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 279 . . . . . . . 8 (𝜑 → (¬ (𝑋𝐼) ∉ ran 𝑌 ↔ ∃𝑗 ∈ (1...𝐾)(𝑌𝑗) = (𝑋𝐼)))
220219con1bid 356 . . . . . . 7 (𝜑 → (¬ ∃𝑗 ∈ (1...𝐾)(𝑌𝑗) = (𝑋𝐼) ↔ (𝑋𝐼) ∉ ran 𝑌))
221214, 220bitrd 279 . . . . . 6 (𝜑 → (∀𝑗 ∈ (1...𝐾) ¬ (𝑌𝑗) = (𝑋𝐼) ↔ (𝑋𝐼) ∉ ran 𝑌))
222221adantr 482 . . . . 5 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → (∀𝑗 ∈ (1...𝐾) ¬ (𝑌𝑗) = (𝑋𝐼) ↔ (𝑋𝐼) ∉ ran 𝑌))
223212, 222mpbid 231 . . . 4 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → (𝑋𝐼) ∉ ran 𝑌)
224 elnelne1 3058 . . . 4 (((𝑋𝐼) ∈ ran 𝑋 ∧ (𝑋𝐼) ∉ ran 𝑌) → ran 𝑋 ≠ ran 𝑌)
22599, 223, 224syl2anc 585 . . 3 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → ran 𝑋 ≠ ran 𝑌)
22645ffund 6722 . . . . . 6 (𝜑 → Fun 𝑌)
227226adantr 482 . . . . 5 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → Fun 𝑌)
22845fdmd 6729 . . . . . . 7 (𝜑 → dom 𝑌 = (1...𝐾))
22970, 228eleqtrrd 2837 . . . . . 6 (𝜑𝐼 ∈ dom 𝑌)
230229adantr 482 . . . . 5 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → 𝐼 ∈ dom 𝑌)
231 fvelrn 7079 . . . . 5 ((Fun 𝑌𝐼 ∈ dom 𝑌) → (𝑌𝐼) ∈ ran 𝑌)
232227, 230, 231syl2anc 585 . . . 4 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → (𝑌𝐼) ∈ ran 𝑌)
2331003ad2ant3 1136 . . . . . . . . . . 11 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝑗 ∈ ℕ)
234233nnred 12227 . . . . . . . . . 10 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝑗 ∈ ℝ)
2351033ad2ant1 1134 . . . . . . . . . 10 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝐼 ∈ ℝ)
236234, 235lttri4d 11355 . . . . . . . . 9 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑗 < 𝐼𝑗 = 𝐼𝐼 < 𝑗))
237313ad2ant1 1134 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝑋:(1...𝐾)⟶(1...𝑁))
238 simp3 1139 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝑗 ∈ (1...𝐾))
239237, 238ffvelcdmd 7088 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝑗) ∈ (1...𝑁))
240109sseli 3979 . . . . . . . . . . . . . 14 ((𝑋𝑗) ∈ (1...𝑁) → (𝑋𝑗) ∈ ℕ)
241239, 240syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝑗) ∈ ℕ)
242241nnred 12227 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝑗) ∈ ℝ)
243242adantr 482 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑋𝑗) ∈ ℝ)
2441883ad2ant1 1134 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)))
245244adantr 482 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)))
246 simpl3 1194 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → 𝑗 ∈ (1...𝐾))
247703ad2ant1 1134 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝐼 ∈ (1...𝐾))
248247adantr 482 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → 𝐼 ∈ (1...𝐾))
249 fveq2 6892 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑗 → (𝑌𝑥) = (𝑌𝑗))
250249breq1d 5159 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑗 → ((𝑌𝑥) < (𝑌𝑦) ↔ (𝑌𝑗) < (𝑌𝑦)))
251121, 250imbi12d 345 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑗 → ((𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)) ↔ (𝑗 < 𝑦 → (𝑌𝑗) < (𝑌𝑦))))
252 fveq2 6892 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝐼 → (𝑌𝑦) = (𝑌𝐼))
253252breq2d 5161 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐼 → ((𝑌𝑗) < (𝑌𝑦) ↔ (𝑌𝑗) < (𝑌𝐼)))
254125, 253imbi12d 345 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐼 → ((𝑗 < 𝑦 → (𝑌𝑗) < (𝑌𝑦)) ↔ (𝑗 < 𝐼 → (𝑌𝑗) < (𝑌𝐼))))
255251, 254rspc2v 3623 . . . . . . . . . . . . . . 15 ((𝑗 ∈ (1...𝐾) ∧ 𝐼 ∈ (1...𝐾)) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)) → (𝑗 < 𝐼 → (𝑌𝑗) < (𝑌𝐼))))
256246, 248, 255syl2anc 585 . . . . . . . . . . . . . 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 840 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑌𝑗) < (𝑌𝐼))
2591693adantl2 1168 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑋𝑗) = (𝑌𝑗))
260259breq1d 5159 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → ((𝑋𝑗) < (𝑌𝐼) ↔ (𝑌𝑗) < (𝑌𝐼)))
261258, 260mpbird 257 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑋𝑗) < (𝑌𝐼))
262243, 261ltned 11350 . . . . . . . . . 10 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑋𝑗) ≠ (𝑌𝐼))
263893ad2ant1 1134 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑌𝐼) ∈ ℝ)
264263adantr 482 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑌𝐼) ∈ ℝ)
265 simpl2 1193 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑌𝐼) < (𝑋𝐼))
266264, 265ltned 11350 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑌𝐼) ≠ (𝑋𝐼))
267266necomd 2997 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑋𝐼) ≠ (𝑌𝐼))
268 fveq2 6892 . . . . . . . . . . . . 13 (𝑗 = 𝐼 → (𝑋𝑗) = (𝑋𝐼))
269268neeq1d 3001 . . . . . . . . . . . 12 (𝑗 = 𝐼 → ((𝑋𝑗) ≠ (𝑌𝐼) ↔ (𝑋𝐼) ≠ (𝑌𝐼)))
270269adantl 483 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → ((𝑋𝑗) ≠ (𝑌𝐼) ↔ (𝑋𝐼) ≠ (𝑌𝐼)))
271267, 270mpbird 257 . . . . . . . . . 10 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑋𝑗) ≠ (𝑌𝐼))
272263adantr 482 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝐼) ∈ ℝ)
273873ad2ant1 1134 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝐼) ∈ ℝ)
274273adantr 482 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝐼) ∈ ℝ)
275242adantr 482 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝑗) ∈ ℝ)
276 simpl2 1193 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝐼) < (𝑋𝐼))
2771153ad2ant1 1134 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)))
278277adantr 482 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)))
279247adantr 482 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → 𝐼 ∈ (1...𝐾))
280238adantr 482 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → 𝑗 ∈ (1...𝐾))
281 fveq2 6892 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝐼 → (𝑋𝑥) = (𝑋𝐼))
282281breq1d 5159 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐼 → ((𝑋𝑥) < (𝑋𝑦) ↔ (𝑋𝐼) < (𝑋𝑦)))
283193, 282imbi12d 345 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐼 → ((𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)) ↔ (𝐼 < 𝑦 → (𝑋𝐼) < (𝑋𝑦))))
284 fveq2 6892 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑗 → (𝑋𝑦) = (𝑋𝑗))
285284breq2d 5161 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑗 → ((𝑋𝐼) < (𝑋𝑦) ↔ (𝑋𝐼) < (𝑋𝑗)))
286197, 285imbi12d 345 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑗 → ((𝐼 < 𝑦 → (𝑋𝐼) < (𝑋𝑦)) ↔ (𝐼 < 𝑗 → (𝑋𝐼) < (𝑋𝑗))))
287283, 286rspc2v 3623 . . . . . . . . . . . . . . . 16 ((𝐼 ∈ (1...𝐾) ∧ 𝑗 ∈ (1...𝐾)) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)) → (𝐼 < 𝑗 → (𝑋𝐼) < (𝑋𝑗))))
288279, 280, 287syl2anc 585 . . . . . . . . . . . . . . 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 840 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝐼) < (𝑋𝑗))
291272, 274, 275, 276, 290lttrd 11375 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝐼) < (𝑋𝑗))
292272, 291ltned 11350 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝐼) ≠ (𝑋𝑗))
293292necomd 2997 . . . . . . . . . 10 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝑗) ≠ (𝑌𝐼))
294262, 271, 2933jaodan 1431 . . . . . . . . 9 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ (𝑗 < 𝐼𝑗 = 𝐼𝐼 < 𝑗)) → (𝑋𝑗) ≠ (𝑌𝐼))
295236, 294mpdan 686 . . . . . . . 8 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝑗) ≠ (𝑌𝐼))
2962953expa 1119 . . . . . . 7 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝑗) ≠ (𝑌𝐼))
297296neneqd 2946 . . . . . 6 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) ∧ 𝑗 ∈ (1...𝐾)) → ¬ (𝑋𝑗) = (𝑌𝐼))
298297ralrimiva 3147 . . . . 5 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → ∀𝑗 ∈ (1...𝐾) ¬ (𝑋𝑗) = (𝑌𝐼))
299 ralnex 3073 . . . . . . . 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 3057 . . . . . . . . . 10 (¬ (𝑌𝐼) ∉ ran 𝑋 ↔ (𝑌𝐼) ∈ ran 𝑋)
302301a1i 11 . . . . . . . . 9 (𝜑 → (¬ (𝑌𝐼) ∉ ran 𝑋 ↔ (𝑌𝐼) ∈ ran 𝑋))
303 fvelrnb 6953 . . . . . . . . . 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 279 . . . . . . . 8 (𝜑 → (¬ (𝑌𝐼) ∉ ran 𝑋 ↔ ∃𝑗 ∈ (1...𝐾)(𝑋𝑗) = (𝑌𝐼)))
306305con1bid 356 . . . . . . 7 (𝜑 → (¬ ∃𝑗 ∈ (1...𝐾)(𝑋𝑗) = (𝑌𝐼) ↔ (𝑌𝐼) ∉ ran 𝑋))
307300, 306bitrd 279 . . . . . 6 (𝜑 → (∀𝑗 ∈ (1...𝐾) ¬ (𝑋𝑗) = (𝑌𝐼) ↔ (𝑌𝐼) ∉ ran 𝑋))
308307adantr 482 . . . . 5 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → (∀𝑗 ∈ (1...𝐾) ¬ (𝑋𝑗) = (𝑌𝐼) ↔ (𝑌𝐼) ∉ ran 𝑋))
309298, 308mpbid 231 . . . 4 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → (𝑌𝐼) ∉ ran 𝑋)
310 elnelne1 3058 . . . . 5 (((𝑌𝐼) ∈ ran 𝑌 ∧ (𝑌𝐼) ∉ ran 𝑋) → ran 𝑌 ≠ ran 𝑋)
311310necomd 2997 . . . 4 (((𝑌𝐼) ∈ ran 𝑌 ∧ (𝑌𝐼) ∉ ran 𝑋) → ran 𝑋 ≠ ran 𝑌)
312232, 309, 311syl2anc 585 . . 3 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → ran 𝑋 ≠ ran 𝑌)
313225, 312jaodan 957 . 2 ((𝜑 ∧ ((𝑋𝐼) < (𝑌𝐼) ∨ (𝑌𝐼) < (𝑋𝐼))) → ran 𝑋 ≠ ran 𝑌)
31492, 313mpdan 686 1 (𝜑 → ran 𝑋 ≠ ran 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846  w3o 1087  w3a 1088  wal 1540   = wceq 1542  wcel 2107  {cab 2710  wne 2941  wnel 3047  wral 3062  wrex 3071  {crab 3433  wss 3949  c0 4323   class class class wbr 5149   Or wor 5588  dom cdm 5677  ran crn 5678  Fun wfun 6538   Fn wfn 6539  wf 6540  cfv 6544  (class class class)co 7409  Fincfn 8939  infcinf 9436  cr 11109  1c1 11111   < clt 11248  cle 11249  cn 12212  0cn0 12472  ...cfz 13484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485
This theorem is referenced by:  sticksstones2  40963
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