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Theorem sticksstones1 42148
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 11342 . . . . . . 7 < Or ℝ
43a1i 11 . . . . . 6 (𝜑 → < Or ℝ)
5 fzfid 14015 . . . . . . . 8 (𝜑 → (1...𝐾) ∈ Fin)
6 ssrab2 4079 . . . . . . . . 9 {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ⊆ (1...𝐾)
76a1i 11 . . . . . . . 8 (𝜑 → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ⊆ (1...𝐾))
8 ssfi 9214 . . . . . . . 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 4387 . . . . . . . . . . . . 13 ({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} = ∅ ↔ ∀𝑧 ∈ (1...𝐾) ¬ (𝑋𝑧) ≠ (𝑌𝑧))
12 nne 2943 . . . . . . . . . . . . . 14 (¬ (𝑋𝑧) ≠ (𝑌𝑧) ↔ (𝑋𝑧) = (𝑌𝑧))
1312ralbii 3092 . . . . . . . . . . . . 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 6715 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑋 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑋:(1...𝐾)⟶(1...𝑁)))
16 fveq1 6904 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 = 𝑋 → (𝑓𝑥) = (𝑋𝑥))
17 fveq1 6904 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 = 𝑋 → (𝑓𝑦) = (𝑋𝑦))
1816, 17breq12d 5155 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑋 → ((𝑓𝑥) < (𝑓𝑦) ↔ (𝑋𝑥) < (𝑋𝑦)))
1918imbi2d 340 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = 𝑋 → ((𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ (𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦))))
20192ralbidv 3220 . . . . . . . . . . . . . . . . . . . . 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 2880 . . . . . . . . . . . . . . . . . . . . . . . . 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 230 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑓(𝑓𝐴 ↔ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦))))
2524spi 2183 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓𝐴 ↔ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦))))
2625biimpi 216 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓𝐴 → (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦))))
2726adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑓𝐴) → (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦))))
2827ralrimiva 3145 . . . . . . . . . . . . . . . . . . . 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 3622 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑋:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦))))
3130simpld 494 . . . . . . . . . . . . . . . . . 18 (𝜑𝑋:(1...𝐾)⟶(1...𝑁))
3231ffnd 6736 . . . . . . . . . . . . . . . . 17 (𝜑𝑋 Fn (1...𝐾))
3332adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧)) → 𝑋 Fn (1...𝐾))
34 sticksstones1.5 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑌𝐴)
35 feq1 6715 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑌 → (𝑓:(1...𝐾)⟶(1...𝑁) ↔ 𝑌:(1...𝐾)⟶(1...𝑁)))
36 fveq1 6904 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = 𝑌 → (𝑓𝑥) = (𝑌𝑥))
37 fveq1 6904 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = 𝑌 → (𝑓𝑦) = (𝑌𝑦))
3836, 37breq12d 5155 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 = 𝑌 → ((𝑓𝑥) < (𝑓𝑦) ↔ (𝑌𝑥) < (𝑌𝑦)))
3938imbi2d 340 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 = 𝑌 → ((𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)) ↔ (𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦))))
40392ralbidv 3220 . . . . . . . . . . . . . . . . . . . . . . 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 3622 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑌:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦))))
4342adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑌𝐴) → (𝑌:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦))))
4434, 43mpdan 687 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑌:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦))))
4544simpld 494 . . . . . . . . . . . . . . . . . 18 (𝜑𝑌:(1...𝐾)⟶(1...𝑁))
4645ffnd 6736 . . . . . . . . . . . . . . . . 17 (𝜑𝑌 Fn (1...𝐾))
4746adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧)) → 𝑌 Fn (1...𝐾))
48 eqfnfv 7050 . . . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . . 14 ((𝜑 ∧ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧)) → (∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧) ↔ 𝑋 = 𝑌))
5150biimpd 229 . . . . . . . . . . . . 13 ((𝜑 ∧ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧)) → (∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧) → 𝑋 = 𝑌))
5251syldbl2 841 . . . . . . . . . . . 12 ((𝜑 ∧ ∀𝑧 ∈ (1...𝐾)(𝑋𝑧) = (𝑌𝑧)) → 𝑋 = 𝑌)
5314, 52sylan2b 594 . . . . . . . . . . 11 ((𝜑 ∧ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} = ∅) → 𝑋 = 𝑌)
5453ex 412 . . . . . . . . . 10 (𝜑 → ({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} = ∅ → 𝑋 = 𝑌))
5554necon3d 2960 . . . . . . . . 9 (𝜑 → (𝑋𝑌 → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ≠ ∅))
5655imp 406 . . . . . . . 8 ((𝜑𝑋𝑌) → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ≠ ∅)
5710, 56mpdan 687 . . . . . . 7 (𝜑 → {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ≠ ∅)
58 fz1ssnn 13596 . . . . . . . . . 10 (1...𝐾) ⊆ ℕ
5958a1i 11 . . . . . . . . 9 (𝜑 → (1...𝐾) ⊆ ℕ)
60 nnssre 12271 . . . . . . . . . 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 1128 . . . . . 6 (𝜑 → ({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ∈ Fin ∧ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ≠ ∅ ∧ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)} ⊆ ℝ))
65 fiinfcl 9542 . . . . . 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 2840 . . . 4 (𝜑𝐼 ∈ {𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)})
687, 66sseldd 3983 . . . . . 6 (𝜑 → inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < ) ∈ (1...𝐾))
692eleq1d 2825 . . . . . 6 (𝜑 → (𝐼 ∈ (1...𝐾) ↔ inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < ) ∈ (1...𝐾)))
7068, 69mpbird 257 . . . . 5 (𝜑𝐼 ∈ (1...𝐾))
71 fveq2 6905 . . . . . . 7 (𝑧 = 𝐼 → (𝑋𝑧) = (𝑋𝐼))
72 fveq2 6905 . . . . . . 7 (𝑧 = 𝐼 → (𝑌𝑧) = (𝑌𝐼))
7371, 72neeq12d 3001 . . . . . 6 (𝑧 = 𝐼 → ((𝑋𝑧) ≠ (𝑌𝑧) ↔ (𝑋𝐼) ≠ (𝑌𝐼)))
7473elrab3 3692 . . . . 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 232 . . 3 (𝜑 → (𝑋𝐼) ≠ (𝑌𝐼))
77 nfv 1913 . . . . . 6 𝑎𝜑
78 nfcv 2904 . . . . . 6 𝑎(1...𝑁)
79 nfcv 2904 . . . . . 6 𝑎
80 elfznn 13594 . . . . . . . . 9 (𝑎 ∈ (1...𝑁) → 𝑎 ∈ ℕ)
8180adantl 481 . . . . . . . 8 ((𝜑𝑎 ∈ (1...𝑁)) → 𝑎 ∈ ℕ)
82 nnre 12274 . . . . . . . 8 (𝑎 ∈ ℕ → 𝑎 ∈ ℝ)
8381, 82syl 17 . . . . . . 7 ((𝜑𝑎 ∈ (1...𝑁)) → 𝑎 ∈ ℝ)
8483ex 412 . . . . . 6 (𝜑 → (𝑎 ∈ (1...𝑁) → 𝑎 ∈ ℝ))
8577, 78, 79, 84ssrd 3987 . . . . 5 (𝜑 → (1...𝑁) ⊆ ℝ)
8631, 70ffvelcdmd 7104 . . . . 5 (𝜑 → (𝑋𝐼) ∈ (1...𝑁))
8785, 86sseldd 3983 . . . 4 (𝜑 → (𝑋𝐼) ∈ ℝ)
8845, 70ffvelcdmd 7104 . . . . 5 (𝜑 → (𝑌𝐼) ∈ (1...𝑁))
8985, 88sseldd 3983 . . . 4 (𝜑 → (𝑌𝐼) ∈ ℝ)
90 lttri2 11344 . . . 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 232 . 2 (𝜑 → ((𝑋𝐼) < (𝑌𝐼) ∨ (𝑌𝐼) < (𝑋𝐼)))
9331ffund 6739 . . . . . 6 (𝜑 → Fun 𝑋)
9493adantr 480 . . . . 5 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → Fun 𝑋)
9531fdmd 6745 . . . . . . 7 (𝜑 → dom 𝑋 = (1...𝐾))
9670, 95eleqtrrd 2843 . . . . . 6 (𝜑𝐼 ∈ dom 𝑋)
9796adantr 480 . . . . 5 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → 𝐼 ∈ dom 𝑋)
98 fvelrn 7095 . . . . 5 ((Fun 𝑋𝐼 ∈ dom 𝑋) → (𝑋𝐼) ∈ ran 𝑋)
9994, 97, 98syl2anc 584 . . . 4 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → (𝑋𝐼) ∈ ran 𝑋)
100 elfznn 13594 . . . . . . . . . . . 12 (𝑗 ∈ (1...𝐾) → 𝑗 ∈ ℕ)
1011003ad2ant3 1135 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝑗 ∈ ℕ)
102101nnred 12282 . . . . . . . . . 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 11403 . . . . . . . . 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 7104 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑌𝑗) ∈ (1...𝑁))
109 fz1ssnn 13596 . . . . . . . . . . . . . . 15 (1...𝑁) ⊆ ℕ
110109sseli 3978 . . . . . . . . . . . . . 14 ((𝑌𝑗) ∈ (1...𝑁) → (𝑌𝑗) ∈ ℕ)
111 nnre 12274 . . . . . . . . . . . . . 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 480 . . . . . . . . . . 11 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑌𝑗) ∈ ℝ)
11530simprd 495 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → 𝐼 ∈ (1...𝐾))
121 breq1 5145 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑗 → (𝑥 < 𝑦𝑗 < 𝑦))
122 fveq2 6905 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑗 → (𝑋𝑥) = (𝑋𝑗))
123122breq1d 5152 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑗 → ((𝑋𝑥) < (𝑋𝑦) ↔ (𝑋𝑗) < (𝑋𝑦)))
124121, 123imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑗 → ((𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)) ↔ (𝑗 < 𝑦 → (𝑋𝑗) < (𝑋𝑦))))
125 breq2 5146 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐼 → (𝑗 < 𝑦𝑗 < 𝐼))
126 fveq2 6905 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝐼 → (𝑋𝑦) = (𝑋𝐼))
127126breq2d 5154 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐼 → ((𝑋𝑗) < (𝑋𝑦) ↔ (𝑋𝑗) < (𝑋𝐼)))
128125, 127imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐼 → ((𝑗 < 𝑦 → (𝑋𝑗) < (𝑋𝑦)) ↔ (𝑗 < 𝐼 → (𝑋𝑗) < (𝑋𝐼))))
129124, 128rspc2v 3632 . . . . . . . . . . . . . . 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 841 . . . . . . . . . . . 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 12282 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → 𝑗 ∈ ℝ)
1371033ad2ant1 1133 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → 𝐼 ∈ ℝ)
138136, 137ltnled 11409 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → (𝑗 < 𝐼 ↔ ¬ 𝐼𝑗))
139134, 138mpbid 232 . . . . . . . . . . . . . . . . . . . . . 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 12255 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . . . . . . . . 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 5152 . . . . . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . . . . . 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 1913 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑧(𝑋𝑗) ≠ (𝑌𝑗)
155 fveq2 6905 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = 𝑗 → (𝑋𝑧) = (𝑋𝑗))
156 fveq2 6905 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = 𝑗 → (𝑌𝑧) = (𝑌𝑗))
157155, 156neeq12d 3001 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 = 𝑗 → ((𝑋𝑧) ≠ (𝑌𝑧) ↔ (𝑋𝑗) ≠ (𝑌𝑗)))
158152, 153, 154, 157elrabf 3687 . . . . . . . . . . . . . . . . . . . . . . 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 983 . . . . . . . . . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → (¬ 𝑗 ∈ (1...𝐾) ∨ ¬ (𝑋𝑗) ≠ (𝑌𝑗)))
163 imor 853 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ (1...𝐾) → ¬ (𝑋𝑗) ≠ (𝑌𝑗)) ↔ (¬ 𝑗 ∈ (1...𝐾) ∨ ¬ (𝑋𝑗) ≠ (𝑌𝑗)))
164162, 163sylibr 234 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) → (𝑗 ∈ (1...𝐾) → ¬ (𝑋𝑗) ≠ (𝑌𝑗)))
165164imp 406 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) ∧ 𝑗 ∈ (1...𝐾)) → ¬ (𝑋𝑗) ≠ (𝑌𝑗))
166 nne 2943 . . . . . . . . . . . . . . . . . 18 (¬ (𝑋𝑗) ≠ (𝑌𝑗) ↔ (𝑋𝑗) = (𝑌𝑗))
167165, 166sylib 218 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (1...𝐾) ∧ 𝑗 < 𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝑗) = (𝑌𝑗))
168133, 167mpdan 687 . . . . . . . . . . . . . . . 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 5152 . . . . . . . . . . . 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 11398 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑌𝑗) ≠ (𝑋𝐼))
175763ad2ant1 1133 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝐼) ≠ (𝑌𝐼))
176175adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑋𝐼) ≠ (𝑌𝐼))
177176necomd 2995 . . . . . . . . . . 11 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑌𝐼) ≠ (𝑋𝐼))
178 fveq2 6905 . . . . . . . . . . . . 13 (𝑗 = 𝐼 → (𝑌𝑗) = (𝑌𝐼))
179178neeq1d 2999 . . . . . . . . . . . 12 (𝑗 = 𝐼 → ((𝑌𝑗) ≠ (𝑋𝐼) ↔ (𝑌𝐼) ≠ (𝑋𝐼)))
180179adantl 481 . . . . . . . . . . 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 1133 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝐼) ∈ ℝ)
183182adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝐼) ∈ ℝ)
184893ad2ant1 1133 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑌𝐼) ∈ ℝ)
185184adantr 480 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝐼) ∈ ℝ)
186113adantr 480 . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)))
191119adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → 𝐼 ∈ (1...𝐾))
192107adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → 𝑗 ∈ (1...𝐾))
193 breq1 5145 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐼 → (𝑥 < 𝑦𝐼 < 𝑦))
194 fveq2 6905 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝐼 → (𝑌𝑥) = (𝑌𝐼))
195194breq1d 5152 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐼 → ((𝑌𝑥) < (𝑌𝑦) ↔ (𝑌𝐼) < (𝑌𝑦)))
196193, 195imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐼 → ((𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)) ↔ (𝐼 < 𝑦 → (𝑌𝐼) < (𝑌𝑦))))
197 breq2 5146 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑗 → (𝐼 < 𝑦𝐼 < 𝑗))
198 fveq2 6905 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑗 → (𝑌𝑦) = (𝑌𝑗))
199198breq2d 5154 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑗 → ((𝑌𝐼) < (𝑌𝑦) ↔ (𝑌𝐼) < (𝑌𝑗)))
200197, 199imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑗 → ((𝐼 < 𝑦 → (𝑌𝐼) < (𝑌𝑦)) ↔ (𝐼 < 𝑗 → (𝑌𝐼) < (𝑌𝑗))))
201196, 200rspc2v 3632 . . . . . . . . . . . . . . . 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 841 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝐼) < (𝑌𝑗))
205183, 185, 186, 187, 204lttrd 11423 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝐼) < (𝑌𝑗))
206183, 205ltned 11398 . . . . . . . . . . 11 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝐼) ≠ (𝑌𝑗))
207206necomd 2995 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝑗) ≠ (𝑋𝐼))
208174, 181, 2073jaodan 1432 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ (𝑗 < 𝐼𝑗 = 𝐼𝐼 < 𝑗)) → (𝑌𝑗) ≠ (𝑋𝐼))
209105, 208mpdan 687 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑌𝑗) ≠ (𝑋𝐼))
2102093expa 1118 . . . . . . 7 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) ∧ 𝑗 ∈ (1...𝐾)) → (𝑌𝑗) ≠ (𝑋𝐼))
211210neneqd 2944 . . . . . 6 (((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) ∧ 𝑗 ∈ (1...𝐾)) → ¬ (𝑌𝑗) = (𝑋𝐼))
212211ralrimiva 3145 . . . . 5 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → ∀𝑗 ∈ (1...𝐾) ¬ (𝑌𝑗) = (𝑋𝐼))
213 ralnex 3071 . . . . . . . 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 3055 . . . . . . . . . 10 (¬ (𝑋𝐼) ∉ ran 𝑌 ↔ (𝑋𝐼) ∈ ran 𝑌)
216215a1i 11 . . . . . . . . 9 (𝜑 → (¬ (𝑋𝐼) ∉ ran 𝑌 ↔ (𝑋𝐼) ∈ ran 𝑌))
217 fvelrnb 6968 . . . . . . . . . 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 355 . . . . . . 7 (𝜑 → (¬ ∃𝑗 ∈ (1...𝐾)(𝑌𝑗) = (𝑋𝐼) ↔ (𝑋𝐼) ∉ ran 𝑌))
221214, 220bitrd 279 . . . . . 6 (𝜑 → (∀𝑗 ∈ (1...𝐾) ¬ (𝑌𝑗) = (𝑋𝐼) ↔ (𝑋𝐼) ∉ ran 𝑌))
222221adantr 480 . . . . 5 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → (∀𝑗 ∈ (1...𝐾) ¬ (𝑌𝑗) = (𝑋𝐼) ↔ (𝑋𝐼) ∉ ran 𝑌))
223212, 222mpbid 232 . . . 4 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → (𝑋𝐼) ∉ ran 𝑌)
224 elnelne1 3056 . . . 4 (((𝑋𝐼) ∈ ran 𝑋 ∧ (𝑋𝐼) ∉ ran 𝑌) → ran 𝑋 ≠ ran 𝑌)
22599, 223, 224syl2anc 584 . . 3 ((𝜑 ∧ (𝑋𝐼) < (𝑌𝐼)) → ran 𝑋 ≠ ran 𝑌)
22645ffund 6739 . . . . . 6 (𝜑 → Fun 𝑌)
227226adantr 480 . . . . 5 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → Fun 𝑌)
22845fdmd 6745 . . . . . . 7 (𝜑 → dom 𝑌 = (1...𝐾))
22970, 228eleqtrrd 2843 . . . . . 6 (𝜑𝐼 ∈ dom 𝑌)
230229adantr 480 . . . . 5 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → 𝐼 ∈ dom 𝑌)
231 fvelrn 7095 . . . . 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 12282 . . . . . . . . . 10 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝑗 ∈ ℝ)
2351033ad2ant1 1133 . . . . . . . . . 10 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → 𝐼 ∈ ℝ)
236234, 235lttri4d 11403 . . . . . . . . 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 7104 . . . . . . . . . . . . . 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 12282 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝑗) ∈ ℝ)
243242adantr 480 . . . . . . . . . . 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 480 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → 𝐼 ∈ (1...𝐾))
249 fveq2 6905 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑗 → (𝑌𝑥) = (𝑌𝑗))
250249breq1d 5152 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑗 → ((𝑌𝑥) < (𝑌𝑦) ↔ (𝑌𝑗) < (𝑌𝑦)))
251121, 250imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑗 → ((𝑥 < 𝑦 → (𝑌𝑥) < (𝑌𝑦)) ↔ (𝑗 < 𝑦 → (𝑌𝑗) < (𝑌𝑦))))
252 fveq2 6905 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝐼 → (𝑌𝑦) = (𝑌𝐼))
253252breq2d 5154 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐼 → ((𝑌𝑗) < (𝑌𝑦) ↔ (𝑌𝑗) < (𝑌𝐼)))
254125, 253imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐼 → ((𝑗 < 𝑦 → (𝑌𝑗) < (𝑌𝑦)) ↔ (𝑗 < 𝐼 → (𝑌𝑗) < (𝑌𝐼))))
255251, 254rspc2v 3632 . . . . . . . . . . . . . . 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 841 . . . . . . . . . . . 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 5152 . . . . . . . . . . . 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 11398 . . . . . . . . . 10 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 < 𝐼) → (𝑋𝑗) ≠ (𝑌𝐼))
263893ad2ant1 1133 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑌𝐼) ∈ ℝ)
264263adantr 480 . . . . . . . . . . . . 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 11398 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑌𝐼) ≠ (𝑋𝐼))
267266necomd 2995 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑗 = 𝐼) → (𝑋𝐼) ≠ (𝑌𝐼))
268 fveq2 6905 . . . . . . . . . . . . 13 (𝑗 = 𝐼 → (𝑋𝑗) = (𝑋𝐼))
269268neeq1d 2999 . . . . . . . . . . . 12 (𝑗 = 𝐼 → ((𝑋𝑗) ≠ (𝑌𝐼) ↔ (𝑋𝐼) ≠ (𝑌𝐼)))
270269adantl 481 . . . . . . . . . . 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 480 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝐼) ∈ ℝ)
273873ad2ant1 1133 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝐼) ∈ ℝ)
274273adantr 480 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝐼) ∈ ℝ)
275242adantr 480 . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)))
279247adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → 𝐼 ∈ (1...𝐾))
280238adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → 𝑗 ∈ (1...𝐾))
281 fveq2 6905 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝐼 → (𝑋𝑥) = (𝑋𝐼))
282281breq1d 5152 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐼 → ((𝑋𝑥) < (𝑋𝑦) ↔ (𝑋𝐼) < (𝑋𝑦)))
283193, 282imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐼 → ((𝑥 < 𝑦 → (𝑋𝑥) < (𝑋𝑦)) ↔ (𝐼 < 𝑦 → (𝑋𝐼) < (𝑋𝑦))))
284 fveq2 6905 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑗 → (𝑋𝑦) = (𝑋𝑗))
285284breq2d 5154 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑗 → ((𝑋𝐼) < (𝑋𝑦) ↔ (𝑋𝐼) < (𝑋𝑗)))
286197, 285imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑗 → ((𝐼 < 𝑦 → (𝑋𝐼) < (𝑋𝑦)) ↔ (𝐼 < 𝑗 → (𝑋𝐼) < (𝑋𝑗))))
287283, 286rspc2v 3632 . . . . . . . . . . . . . . . 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 841 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝐼) < (𝑋𝑗))
291272, 274, 275, 276, 290lttrd 11423 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝐼) < (𝑋𝑗))
292272, 291ltned 11398 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑌𝐼) ≠ (𝑋𝑗))
293292necomd 2995 . . . . . . . . . 10 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝐼 < 𝑗) → (𝑋𝑗) ≠ (𝑌𝐼))
294262, 271, 2933jaodan 1432 . . . . . . . . 9 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) ∧ (𝑗 < 𝐼𝑗 = 𝐼𝐼 < 𝑗)) → (𝑋𝑗) ≠ (𝑌𝐼))
295236, 294mpdan 687 . . . . . . . 8 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝑗) ≠ (𝑌𝐼))
2962953expa 1118 . . . . . . 7 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) ∧ 𝑗 ∈ (1...𝐾)) → (𝑋𝑗) ≠ (𝑌𝐼))
297296neneqd 2944 . . . . . 6 (((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) ∧ 𝑗 ∈ (1...𝐾)) → ¬ (𝑋𝑗) = (𝑌𝐼))
298297ralrimiva 3145 . . . . 5 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → ∀𝑗 ∈ (1...𝐾) ¬ (𝑋𝑗) = (𝑌𝐼))
299 ralnex 3071 . . . . . . . 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 3055 . . . . . . . . . 10 (¬ (𝑌𝐼) ∉ ran 𝑋 ↔ (𝑌𝐼) ∈ ran 𝑋)
302301a1i 11 . . . . . . . . 9 (𝜑 → (¬ (𝑌𝐼) ∉ ran 𝑋 ↔ (𝑌𝐼) ∈ ran 𝑋))
303 fvelrnb 6968 . . . . . . . . . 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 355 . . . . . . 7 (𝜑 → (¬ ∃𝑗 ∈ (1...𝐾)(𝑋𝑗) = (𝑌𝐼) ↔ (𝑌𝐼) ∉ ran 𝑋))
307300, 306bitrd 279 . . . . . 6 (𝜑 → (∀𝑗 ∈ (1...𝐾) ¬ (𝑋𝑗) = (𝑌𝐼) ↔ (𝑌𝐼) ∉ ran 𝑋))
308307adantr 480 . . . . 5 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → (∀𝑗 ∈ (1...𝐾) ¬ (𝑋𝑗) = (𝑌𝐼) ↔ (𝑌𝐼) ∉ ran 𝑋))
309298, 308mpbid 232 . . . 4 ((𝜑 ∧ (𝑌𝐼) < (𝑋𝐼)) → (𝑌𝐼) ∉ ran 𝑋)
310 elnelne1 3056 . . . . 5 (((𝑌𝐼) ∈ ran 𝑌 ∧ (𝑌𝐼) ∉ ran 𝑋) → ran 𝑌 ≠ ran 𝑋)
311310necomd 2995 . . . 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 959 . 2 ((𝜑 ∧ ((𝑋𝐼) < (𝑌𝐼) ∨ (𝑌𝐼) < (𝑋𝐼))) → ran 𝑋 ≠ ran 𝑌)
31492, 313mpdan 687 1 (𝜑 → ran 𝑋 ≠ ran 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3o 1085  w3a 1086  wal 1537   = wceq 1539  wcel 2107  {cab 2713  wne 2939  wnel 3045  wral 3060  wrex 3069  {crab 3435  wss 3950  c0 4332   class class class wbr 5142   Or wor 5590  dom cdm 5684  ran crn 5685  Fun wfun 6554   Fn wfn 6555  wf 6556  cfv 6560  (class class class)co 7432  Fincfn 8986  infcinf 9482  cr 11155  1c1 11157   < clt 11296  cle 11297  cn 12267  0cn0 12528  ...cfz 13548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-inf 9484  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-n0 12529  df-z 12616  df-uz 12880  df-fz 13549
This theorem is referenced by:  sticksstones2  42149
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