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Theorem vdwlem12 16987
Description: Lemma for vdw 16989. 𝐾 = 2 base case of induction. (Contributed by Mario Carneiro, 18-Aug-2014.)
Hypotheses
Ref Expression
vdw.r (𝜑𝑅 ∈ Fin)
vdwlem12.f (𝜑𝐹:(1...((♯‘𝑅) + 1))⟶𝑅)
vdwlem12.2 (𝜑 → ¬ 2 MonoAP 𝐹)
Assertion
Ref Expression
vdwlem12 ¬ 𝜑

Proof of Theorem vdwlem12
Dummy variables 𝑎 𝑐 𝑑 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vdw.r . . . . . . 7 (𝜑𝑅 ∈ Fin)
2 hashcl 14366 . . . . . . 7 (𝑅 ∈ Fin → (♯‘𝑅) ∈ ℕ0)
31, 2syl 17 . . . . . 6 (𝜑 → (♯‘𝑅) ∈ ℕ0)
43nn0red 12577 . . . . 5 (𝜑 → (♯‘𝑅) ∈ ℝ)
54ltp1d 12188 . . . 4 (𝜑 → (♯‘𝑅) < ((♯‘𝑅) + 1))
6 nn0p1nn 12555 . . . . . . 7 ((♯‘𝑅) ∈ ℕ0 → ((♯‘𝑅) + 1) ∈ ℕ)
73, 6syl 17 . . . . . 6 (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ)
87nnnn0d 12576 . . . . 5 (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ0)
9 hashfz1 14356 . . . . 5 (((♯‘𝑅) + 1) ∈ ℕ0 → (♯‘(1...((♯‘𝑅) + 1))) = ((♯‘𝑅) + 1))
108, 9syl 17 . . . 4 (𝜑 → (♯‘(1...((♯‘𝑅) + 1))) = ((♯‘𝑅) + 1))
115, 10breqtrrd 5172 . . 3 (𝜑 → (♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))))
12 fzfi 13984 . . . 4 (1...((♯‘𝑅) + 1)) ∈ Fin
13 hashsdom 14391 . . . 4 ((𝑅 ∈ Fin ∧ (1...((♯‘𝑅) + 1)) ∈ Fin) → ((♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))) ↔ 𝑅 ≺ (1...((♯‘𝑅) + 1))))
141, 12, 13sylancl 584 . . 3 (𝜑 → ((♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))) ↔ 𝑅 ≺ (1...((♯‘𝑅) + 1))))
1511, 14mpbid 231 . 2 (𝜑𝑅 ≺ (1...((♯‘𝑅) + 1)))
16 vdwlem12.f . . . . 5 (𝜑𝐹:(1...((♯‘𝑅) + 1))⟶𝑅)
17 fveq2 6891 . . . . . . . . 9 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
18 fveq2 6891 . . . . . . . . 9 (𝑤 = 𝑦 → (𝐹𝑤) = (𝐹𝑦))
1917, 18eqeqan12d 2740 . . . . . . . 8 ((𝑧 = 𝑥𝑤 = 𝑦) → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑥) = (𝐹𝑦)))
20 eqeq12 2743 . . . . . . . 8 ((𝑧 = 𝑥𝑤 = 𝑦) → (𝑧 = 𝑤𝑥 = 𝑦))
2119, 20imbi12d 343 . . . . . . 7 ((𝑧 = 𝑥𝑤 = 𝑦) → (((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
22 fveq2 6891 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
23 fveq2 6891 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
2422, 23eqeqan12d 2740 . . . . . . . . 9 ((𝑧 = 𝑦𝑤 = 𝑥) → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑦) = (𝐹𝑥)))
25 eqcom 2733 . . . . . . . . 9 ((𝐹𝑦) = (𝐹𝑥) ↔ (𝐹𝑥) = (𝐹𝑦))
2624, 25bitrdi 286 . . . . . . . 8 ((𝑧 = 𝑦𝑤 = 𝑥) → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑥) = (𝐹𝑦)))
27 eqeq12 2743 . . . . . . . . 9 ((𝑧 = 𝑦𝑤 = 𝑥) → (𝑧 = 𝑤𝑦 = 𝑥))
28 eqcom 2733 . . . . . . . . 9 (𝑦 = 𝑥𝑥 = 𝑦)
2927, 28bitrdi 286 . . . . . . . 8 ((𝑧 = 𝑦𝑤 = 𝑥) → (𝑧 = 𝑤𝑥 = 𝑦))
3026, 29imbi12d 343 . . . . . . 7 ((𝑧 = 𝑦𝑤 = 𝑥) → (((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
31 elfznn 13576 . . . . . . . . . 10 (𝑥 ∈ (1...((♯‘𝑅) + 1)) → 𝑥 ∈ ℕ)
3231nnred 12271 . . . . . . . . 9 (𝑥 ∈ (1...((♯‘𝑅) + 1)) → 𝑥 ∈ ℝ)
3332ssriv 3983 . . . . . . . 8 (1...((♯‘𝑅) + 1)) ⊆ ℝ
3433a1i 11 . . . . . . 7 (𝜑 → (1...((♯‘𝑅) + 1)) ⊆ ℝ)
35 biidd 261 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
36 simplr3 1214 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥𝑦)
37 vdwlem12.2 . . . . . . . . . . 11 (𝜑 → ¬ 2 MonoAP 𝐹)
3837ad2antrr 724 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → ¬ 2 MonoAP 𝐹)
39 3simpa 1145 . . . . . . . . . . . 12 ((𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦) → (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1))))
40 simplrl 775 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ (1...((♯‘𝑅) + 1)))
4140, 31syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℕ)
42 simprr 771 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 < 𝑦)
43 simplrr 776 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ (1...((♯‘𝑅) + 1)))
44 elfznn 13576 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1...((♯‘𝑅) + 1)) → 𝑦 ∈ ℕ)
4543, 44syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℕ)
46 nnsub 12300 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 < 𝑦 ↔ (𝑦𝑥) ∈ ℕ))
4741, 45, 46syl2anc 582 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 < 𝑦 ↔ (𝑦𝑥) ∈ ℕ))
4842, 47mpbid 231 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑦𝑥) ∈ ℕ)
49 df-2 12319 . . . . . . . . . . . . . . . . . . 19 2 = (1 + 1)
5049fveq2i 6894 . . . . . . . . . . . . . . . . . 18 (AP‘2) = (AP‘(1 + 1))
5150oveqi 7427 . . . . . . . . . . . . . . . . 17 (𝑥(AP‘2)(𝑦𝑥)) = (𝑥(AP‘(1 + 1))(𝑦𝑥))
52 1nn0 12532 . . . . . . . . . . . . . . . . . 18 1 ∈ ℕ0
53 vdwapun 16969 . . . . . . . . . . . . . . . . . 18 ((1 ∈ ℕ0𝑥 ∈ ℕ ∧ (𝑦𝑥) ∈ ℕ) → (𝑥(AP‘(1 + 1))(𝑦𝑥)) = ({𝑥} ∪ ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥))))
5452, 41, 48, 53mp3an2i 1463 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥(AP‘(1 + 1))(𝑦𝑥)) = ({𝑥} ∪ ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥))))
5551, 54eqtrid 2778 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥(AP‘2)(𝑦𝑥)) = ({𝑥} ∪ ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥))))
56 simprl 769 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝐹𝑥) = (𝐹𝑦))
5716ad2antrr 724 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝐹:(1...((♯‘𝑅) + 1))⟶𝑅)
5857ffnd 6719 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝐹 Fn (1...((♯‘𝑅) + 1)))
59 fniniseg 7063 . . . . . . . . . . . . . . . . . . . 20 (𝐹 Fn (1...((♯‘𝑅) + 1)) → (𝑥 ∈ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ (𝐹𝑥) = (𝐹𝑦))))
6058, 59syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 ∈ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ (𝐹𝑥) = (𝐹𝑦))))
6140, 56, 60mpbir2and 711 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ (𝐹 “ {(𝐹𝑦)}))
6261snssd 4809 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → {𝑥} ⊆ (𝐹 “ {(𝐹𝑦)}))
6341nncnd 12272 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℂ)
6445nncnd 12272 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℂ)
6563, 64pncan3d 11613 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 + (𝑦𝑥)) = 𝑦)
6665oveq1d 7429 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥)) = (𝑦(AP‘1)(𝑦𝑥)))
67 vdwap1 16972 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℕ ∧ (𝑦𝑥) ∈ ℕ) → (𝑦(AP‘1)(𝑦𝑥)) = {𝑦})
6845, 48, 67syl2anc 582 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑦(AP‘1)(𝑦𝑥)) = {𝑦})
6966, 68eqtrd 2766 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥)) = {𝑦})
70 eqidd 2727 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝐹𝑦) = (𝐹𝑦))
71 fniniseg 7063 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 Fn (1...((♯‘𝑅) + 1)) → (𝑦 ∈ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ (𝐹𝑦) = (𝐹𝑦))))
7258, 71syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑦 ∈ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ (𝐹𝑦) = (𝐹𝑦))))
7343, 70, 72mpbir2and 711 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ (𝐹 “ {(𝐹𝑦)}))
7473snssd 4809 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → {𝑦} ⊆ (𝐹 “ {(𝐹𝑦)}))
7569, 74eqsstrd 4018 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥)) ⊆ (𝐹 “ {(𝐹𝑦)}))
7662, 75unssd 4185 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ({𝑥} ∪ ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥))) ⊆ (𝐹 “ {(𝐹𝑦)}))
7755, 76eqsstrd 4018 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥(AP‘2)(𝑦𝑥)) ⊆ (𝐹 “ {(𝐹𝑦)}))
78 oveq1 7421 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑥 → (𝑎(AP‘2)𝑑) = (𝑥(AP‘2)𝑑))
7978sseq1d 4011 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → ((𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑥(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)})))
80 oveq2 7422 . . . . . . . . . . . . . . . . 17 (𝑑 = (𝑦𝑥) → (𝑥(AP‘2)𝑑) = (𝑥(AP‘2)(𝑦𝑥)))
8180sseq1d 4011 . . . . . . . . . . . . . . . 16 (𝑑 = (𝑦𝑥) → ((𝑥(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑥(AP‘2)(𝑦𝑥)) ⊆ (𝐹 “ {(𝐹𝑦)})))
8279, 81rspc2ev 3621 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℕ ∧ (𝑦𝑥) ∈ ℕ ∧ (𝑥(AP‘2)(𝑦𝑥)) ⊆ (𝐹 “ {(𝐹𝑦)})) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}))
8341, 48, 77, 82syl3anc 1368 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}))
84 fvex 6904 . . . . . . . . . . . . . . 15 (𝐹𝑦) ∈ V
85 sneq 4634 . . . . . . . . . . . . . . . . . 18 (𝑐 = (𝐹𝑦) → {𝑐} = {(𝐹𝑦)})
8685imaeq2d 6060 . . . . . . . . . . . . . . . . 17 (𝑐 = (𝐹𝑦) → (𝐹 “ {𝑐}) = (𝐹 “ {(𝐹𝑦)}))
8786sseq2d 4012 . . . . . . . . . . . . . . . 16 (𝑐 = (𝐹𝑦) → ((𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)})))
88872rexbidv 3210 . . . . . . . . . . . . . . 15 (𝑐 = (𝐹𝑦) → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)})))
8984, 88spcev 3592 . . . . . . . . . . . . . 14 (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}) → ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {𝑐}))
9083, 89syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {𝑐}))
91 ovex 7447 . . . . . . . . . . . . . 14 (1...((♯‘𝑅) + 1)) ∈ V
92 2nn0 12533 . . . . . . . . . . . . . . 15 2 ∈ ℕ0
9392a1i 11 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 2 ∈ ℕ0)
9491, 93, 57vdwmc 16973 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (2 MonoAP 𝐹 ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {𝑐})))
9590, 94mpbird 256 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 2 MonoAP 𝐹)
9639, 95sylanl2 679 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 2 MonoAP 𝐹)
9796expr 455 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝑥 < 𝑦 → 2 MonoAP 𝐹))
9838, 97mtod 197 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → ¬ 𝑥 < 𝑦)
99 simplr1 1212 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥 ∈ (1...((♯‘𝑅) + 1)))
10099, 32syl 17 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥 ∈ ℝ)
101 simplr2 1213 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑦 ∈ (1...((♯‘𝑅) + 1)))
10233, 101sselid 3977 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑦 ∈ ℝ)
103100, 102eqleltd 11397 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝑥 = 𝑦 ↔ (𝑥𝑦 ∧ ¬ 𝑥 < 𝑦)))
10436, 98, 103mpbir2and 711 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥 = 𝑦)
105104ex 411 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
10621, 30, 34, 35, 105wlogle 11786 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
107106ralrimivva 3191 . . . . 5 (𝜑 → ∀𝑥 ∈ (1...((♯‘𝑅) + 1))∀𝑦 ∈ (1...((♯‘𝑅) + 1))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
108 dff13 7260 . . . . 5 (𝐹:(1...((♯‘𝑅) + 1))–1-1𝑅 ↔ (𝐹:(1...((♯‘𝑅) + 1))⟶𝑅 ∧ ∀𝑥 ∈ (1...((♯‘𝑅) + 1))∀𝑦 ∈ (1...((♯‘𝑅) + 1))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
10916, 107, 108sylanbrc 581 . . . 4 (𝜑𝐹:(1...((♯‘𝑅) + 1))–1-1𝑅)
110 f1domg 8993 . . . 4 (𝑅 ∈ Fin → (𝐹:(1...((♯‘𝑅) + 1))–1-1𝑅 → (1...((♯‘𝑅) + 1)) ≼ 𝑅))
1111, 109, 110sylc 65 . . 3 (𝜑 → (1...((♯‘𝑅) + 1)) ≼ 𝑅)
112 domnsym 9127 . . 3 ((1...((♯‘𝑅) + 1)) ≼ 𝑅 → ¬ 𝑅 ≺ (1...((♯‘𝑅) + 1)))
113111, 112syl 17 . 2 (𝜑 → ¬ 𝑅 ≺ (1...((♯‘𝑅) + 1)))
11415, 113pm2.65i 193 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wex 1774  wcel 2099  wral 3051  wrex 3060  cun 3945  wss 3947  {csn 4624   class class class wbr 5144  ccnv 5672  cima 5676   Fn wfn 6539  wf 6540  1-1wf1 6541  cfv 6544  (class class class)co 7414  cdom 8962  csdm 8963  Fincfn 8964  cr 11146  1c1 11148   + caddc 11150   < clt 11287  cle 11288  cmin 11483  cn 12256  2c2 12311  0cn0 12516  ...cfz 13530  chash 14340  APcvdwa 16960   MonoAP cvdwm 16961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736  ax-cnex 11203  ax-resscn 11204  ax-1cn 11205  ax-icn 11206  ax-addcl 11207  ax-addrcl 11208  ax-mulcl 11209  ax-mulrcl 11210  ax-mulcom 11211  ax-addass 11212  ax-mulass 11213  ax-distr 11214  ax-i2m1 11215  ax-1ne0 11216  ax-1rid 11217  ax-rnegex 11218  ax-rrecex 11219  ax-cnre 11220  ax-pre-lttri 11221  ax-pre-lttrn 11222  ax-pre-ltadd 11223  ax-pre-mulgt0 11224
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-int 4948  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6303  df-ord 6369  df-on 6370  df-lim 6371  df-suc 6372  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 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-oadd 8490  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-card 9973  df-pnf 11289  df-mnf 11290  df-xr 11291  df-ltxr 11292  df-le 11293  df-sub 11485  df-neg 11486  df-nn 12257  df-2 12319  df-n0 12517  df-xnn0 12589  df-z 12603  df-uz 12867  df-fz 13531  df-hash 14341  df-vdwap 16963  df-vdwmc 16964
This theorem is referenced by:  vdwlem13  16988
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