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Theorem vdwlem12 15743
 Description: Lemma for vdw 15745. 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 13185 . . . . . . 7 (𝑅 ∈ Fin → (#‘𝑅) ∈ ℕ0)
31, 2syl 17 . . . . . 6 (𝜑 → (#‘𝑅) ∈ ℕ0)
43nn0red 11390 . . . . 5 (𝜑 → (#‘𝑅) ∈ ℝ)
54ltp1d 10992 . . . 4 (𝜑 → (#‘𝑅) < ((#‘𝑅) + 1))
6 nn0p1nn 11370 . . . . . . 7 ((#‘𝑅) ∈ ℕ0 → ((#‘𝑅) + 1) ∈ ℕ)
73, 6syl 17 . . . . . 6 (𝜑 → ((#‘𝑅) + 1) ∈ ℕ)
87nnnn0d 11389 . . . . 5 (𝜑 → ((#‘𝑅) + 1) ∈ ℕ0)
9 hashfz1 13174 . . . . 5 (((#‘𝑅) + 1) ∈ ℕ0 → (#‘(1...((#‘𝑅) + 1))) = ((#‘𝑅) + 1))
108, 9syl 17 . . . 4 (𝜑 → (#‘(1...((#‘𝑅) + 1))) = ((#‘𝑅) + 1))
115, 10breqtrrd 4713 . . 3 (𝜑 → (#‘𝑅) < (#‘(1...((#‘𝑅) + 1))))
12 fzfi 12811 . . . 4 (1...((#‘𝑅) + 1)) ∈ Fin
13 hashsdom 13208 . . . 4 ((𝑅 ∈ Fin ∧ (1...((#‘𝑅) + 1)) ∈ Fin) → ((#‘𝑅) < (#‘(1...((#‘𝑅) + 1))) ↔ 𝑅 ≺ (1...((#‘𝑅) + 1))))
141, 12, 13sylancl 695 . . 3 (𝜑 → ((#‘𝑅) < (#‘(1...((#‘𝑅) + 1))) ↔ 𝑅 ≺ (1...((#‘𝑅) + 1))))
1511, 14mpbid 222 . 2 (𝜑𝑅 ≺ (1...((#‘𝑅) + 1)))
16 vdwlem12.f . . . . 5 (𝜑𝐹:(1...((#‘𝑅) + 1))⟶𝑅)
17 fveq2 6229 . . . . . . . . 9 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
18 fveq2 6229 . . . . . . . . 9 (𝑤 = 𝑦 → (𝐹𝑤) = (𝐹𝑦))
1917, 18eqeqan12d 2667 . . . . . . . 8 ((𝑧 = 𝑥𝑤 = 𝑦) → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑥) = (𝐹𝑦)))
20 eqeq12 2664 . . . . . . . 8 ((𝑧 = 𝑥𝑤 = 𝑦) → (𝑧 = 𝑤𝑥 = 𝑦))
2119, 20imbi12d 333 . . . . . . 7 ((𝑧 = 𝑥𝑤 = 𝑦) → (((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
22 fveq2 6229 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
23 fveq2 6229 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
2422, 23eqeqan12d 2667 . . . . . . . . 9 ((𝑧 = 𝑦𝑤 = 𝑥) → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑦) = (𝐹𝑥)))
25 eqcom 2658 . . . . . . . . 9 ((𝐹𝑦) = (𝐹𝑥) ↔ (𝐹𝑥) = (𝐹𝑦))
2624, 25syl6bb 276 . . . . . . . 8 ((𝑧 = 𝑦𝑤 = 𝑥) → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑥) = (𝐹𝑦)))
27 eqeq12 2664 . . . . . . . . 9 ((𝑧 = 𝑦𝑤 = 𝑥) → (𝑧 = 𝑤𝑦 = 𝑥))
28 eqcom 2658 . . . . . . . . 9 (𝑦 = 𝑥𝑥 = 𝑦)
2927, 28syl6bb 276 . . . . . . . 8 ((𝑧 = 𝑦𝑤 = 𝑥) → (𝑧 = 𝑤𝑥 = 𝑦))
3026, 29imbi12d 333 . . . . . . 7 ((𝑧 = 𝑦𝑤 = 𝑥) → (((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
31 elfznn 12408 . . . . . . . . . 10 (𝑥 ∈ (1...((#‘𝑅) + 1)) → 𝑥 ∈ ℕ)
3231nnred 11073 . . . . . . . . 9 (𝑥 ∈ (1...((#‘𝑅) + 1)) → 𝑥 ∈ ℝ)
3332ssriv 3640 . . . . . . . 8 (1...((#‘𝑅) + 1)) ⊆ ℝ
3433a1i 11 . . . . . . 7 (𝜑 → (1...((#‘𝑅) + 1)) ⊆ ℝ)
35 biidd 252 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
36 simplr3 1125 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥𝑦)
37 vdwlem12.2 . . . . . . . . . . 11 (𝜑 → ¬ 2 MonoAP 𝐹)
3837ad2antrr 762 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → ¬ 2 MonoAP 𝐹)
39 3simpa 1078 . . . . . . . . . . . 12 ((𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥𝑦) → (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1))))
40 simplrl 817 . . . . . . . . . . . . . . . 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 811 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 < 𝑦)
43 simplrr 818 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ (1...((#‘𝑅) + 1)))
44 elfznn 12408 . . . . . . . . . . . . . . . . . 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 11097 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 < 𝑦 ↔ (𝑦𝑥) ∈ ℕ))
4741, 45, 46syl2anc 694 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 < 𝑦 ↔ (𝑦𝑥) ∈ ℕ))
4842, 47mpbid 222 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑦𝑥) ∈ ℕ)
49 df-2 11117 . . . . . . . . . . . . . . . . . . 19 2 = (1 + 1)
5049fveq2i 6232 . . . . . . . . . . . . . . . . . 18 (AP‘2) = (AP‘(1 + 1))
5150oveqi 6703 . . . . . . . . . . . . . . . . 17 (𝑥(AP‘2)(𝑦𝑥)) = (𝑥(AP‘(1 + 1))(𝑦𝑥))
52 1nn0 11346 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℕ0
5352a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 1 ∈ ℕ0)
54 vdwapun 15725 . . . . . . . . . . . . . . . . . 18 ((1 ∈ ℕ0𝑥 ∈ ℕ ∧ (𝑦𝑥) ∈ ℕ) → (𝑥(AP‘(1 + 1))(𝑦𝑥)) = ({𝑥} ∪ ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥))))
5553, 41, 48, 54syl3anc 1366 . . . . . . . . . . . . . . . . 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))))
5651, 55syl5eq 2697 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥(AP‘2)(𝑦𝑥)) = ({𝑥} ∪ ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥))))
57 simprl 809 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝐹𝑥) = (𝐹𝑦))
5816ad2antrr 762 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝐹:(1...((#‘𝑅) + 1))⟶𝑅)
59 ffn 6083 . . . . . . . . . . . . . . . . . . . . 21 (𝐹:(1...((#‘𝑅) + 1))⟶𝑅𝐹 Fn (1...((#‘𝑅) + 1)))
6058, 59syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝐹 Fn (1...((#‘𝑅) + 1)))
61 fniniseg 6378 . . . . . . . . . . . . . . . . . . . 20 (𝐹 Fn (1...((#‘𝑅) + 1)) → (𝑥 ∈ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ (𝐹𝑥) = (𝐹𝑦))))
6260, 61syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 ∈ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ (𝐹𝑥) = (𝐹𝑦))))
6340, 57, 62mpbir2and 977 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ (𝐹 “ {(𝐹𝑦)}))
6463snssd 4372 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → {𝑥} ⊆ (𝐹 “ {(𝐹𝑦)}))
6541nncnd 11074 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℂ)
6645nncnd 11074 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℂ)
6765, 66pncan3d 10433 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 + (𝑦𝑥)) = 𝑦)
6867oveq1d 6705 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥)) = (𝑦(AP‘1)(𝑦𝑥)))
69 vdwap1 15728 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℕ ∧ (𝑦𝑥) ∈ ℕ) → (𝑦(AP‘1)(𝑦𝑥)) = {𝑦})
7045, 48, 69syl2anc 694 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑦(AP‘1)(𝑦𝑥)) = {𝑦})
7168, 70eqtrd 2685 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥)) = {𝑦})
72 eqidd 2652 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝐹𝑦) = (𝐹𝑦))
73 fniniseg 6378 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 Fn (1...((#‘𝑅) + 1)) → (𝑦 ∈ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ (𝐹𝑦) = (𝐹𝑦))))
7460, 73syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑦 ∈ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ (𝐹𝑦) = (𝐹𝑦))))
7543, 72, 74mpbir2and 977 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ (𝐹 “ {(𝐹𝑦)}))
7675snssd 4372 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → {𝑦} ⊆ (𝐹 “ {(𝐹𝑦)}))
7771, 76eqsstrd 3672 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥)) ⊆ (𝐹 “ {(𝐹𝑦)}))
7864, 77unssd 3822 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ({𝑥} ∪ ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥))) ⊆ (𝐹 “ {(𝐹𝑦)}))
7956, 78eqsstrd 3672 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥(AP‘2)(𝑦𝑥)) ⊆ (𝐹 “ {(𝐹𝑦)}))
80 oveq1 6697 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑥 → (𝑎(AP‘2)𝑑) = (𝑥(AP‘2)𝑑))
8180sseq1d 3665 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → ((𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑥(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)})))
82 oveq2 6698 . . . . . . . . . . . . . . . . 17 (𝑑 = (𝑦𝑥) → (𝑥(AP‘2)𝑑) = (𝑥(AP‘2)(𝑦𝑥)))
8382sseq1d 3665 . . . . . . . . . . . . . . . 16 (𝑑 = (𝑦𝑥) → ((𝑥(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑥(AP‘2)(𝑦𝑥)) ⊆ (𝐹 “ {(𝐹𝑦)})))
8481, 83rspc2ev 3355 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℕ ∧ (𝑦𝑥) ∈ ℕ ∧ (𝑥(AP‘2)(𝑦𝑥)) ⊆ (𝐹 “ {(𝐹𝑦)})) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}))
8541, 48, 79, 84syl3anc 1366 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}))
86 fvex 6239 . . . . . . . . . . . . . . 15 (𝐹𝑦) ∈ V
87 sneq 4220 . . . . . . . . . . . . . . . . . 18 (𝑐 = (𝐹𝑦) → {𝑐} = {(𝐹𝑦)})
8887imaeq2d 5501 . . . . . . . . . . . . . . . . 17 (𝑐 = (𝐹𝑦) → (𝐹 “ {𝑐}) = (𝐹 “ {(𝐹𝑦)}))
8988sseq2d 3666 . . . . . . . . . . . . . . . 16 (𝑐 = (𝐹𝑦) → ((𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)})))
90892rexbidv 3086 . . . . . . . . . . . . . . 15 (𝑐 = (𝐹𝑦) → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)})))
9186, 90spcev 3331 . . . . . . . . . . . . . 14 (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}) → ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {𝑐}))
9285, 91syl 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}))
93 ovex 6718 . . . . . . . . . . . . . 14 (1...((#‘𝑅) + 1)) ∈ V
94 2nn0 11347 . . . . . . . . . . . . . . 15 2 ∈ ℕ0
9594a1i 11 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 2 ∈ ℕ0)
9693, 95, 58vdwmc 15729 . . . . . . . . . . . . 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})))
9792, 96mpbird 247 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 2 MonoAP 𝐹)
9839, 97sylanl2 684 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 2 MonoAP 𝐹)
9998expr 642 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝑥 < 𝑦 → 2 MonoAP 𝐹))
10038, 99mtod 189 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → ¬ 𝑥 < 𝑦)
101 simplr1 1123 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥 ∈ (1...((#‘𝑅) + 1)))
102101, 32syl 17 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥 ∈ ℝ)
103 simplr2 1124 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑦 ∈ (1...((#‘𝑅) + 1)))
10433, 103sseldi 3634 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑦 ∈ ℝ)
105102, 104eqleltd 10219 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝑥 = 𝑦 ↔ (𝑥𝑦 ∧ ¬ 𝑥 < 𝑦)))
10636, 100, 105mpbir2and 977 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥 = 𝑦)
107106ex 449 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥𝑦)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
10821, 30, 34, 35, 107wlogle 10599 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
109108ralrimivva 3000 . . . . 5 (𝜑 → ∀𝑥 ∈ (1...((#‘𝑅) + 1))∀𝑦 ∈ (1...((#‘𝑅) + 1))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
110 dff13 6552 . . . . 5 (𝐹:(1...((#‘𝑅) + 1))–1-1𝑅 ↔ (𝐹:(1...((#‘𝑅) + 1))⟶𝑅 ∧ ∀𝑥 ∈ (1...((#‘𝑅) + 1))∀𝑦 ∈ (1...((#‘𝑅) + 1))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
11116, 109, 110sylanbrc 699 . . . 4 (𝜑𝐹:(1...((#‘𝑅) + 1))–1-1𝑅)
112 f1domg 8017 . . . 4 (𝑅 ∈ Fin → (𝐹:(1...((#‘𝑅) + 1))–1-1𝑅 → (1...((#‘𝑅) + 1)) ≼ 𝑅))
1131, 111, 112sylc 65 . . 3 (𝜑 → (1...((#‘𝑅) + 1)) ≼ 𝑅)
114 domnsym 8127 . . 3 ((1...((#‘𝑅) + 1)) ≼ 𝑅 → ¬ 𝑅 ≺ (1...((#‘𝑅) + 1)))
115113, 114syl 17 . 2 (𝜑 → ¬ 𝑅 ≺ (1...((#‘𝑅) + 1)))
11615, 115pm2.65i 185 1 ¬ 𝜑
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523  ∃wex 1744   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942   ∪ cun 3605   ⊆ wss 3607  {csn 4210   class class class wbr 4685  ◡ccnv 5142   “ cima 5146   Fn wfn 5921  ⟶wf 5922  –1-1→wf1 5923  ‘cfv 5926  (class class class)co 6690   ≼ cdom 7995   ≺ csdm 7996  Fincfn 7997  ℝcr 9973  1c1 9975   + caddc 9977   < clt 10112   ≤ cle 10113   − cmin 10304  ℕcn 11058  2c2 11108  ℕ0cn0 11330  ...cfz 12364  #chash 13157  APcvdwa 15716   MonoAP cvdwm 15717 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158  df-vdwap 15719  df-vdwmc 15720 This theorem is referenced by:  vdwlem13  15744
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