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Theorem unxpwdom2 8534
Description: Lemma for unxpwdom 8535. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
unxpwdom2 ((𝐴 × 𝐴) ≈ (𝐵𝐶) → (𝐴* 𝐵𝐴𝐶))

Proof of Theorem unxpwdom2
Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ensym 8046 . 2 ((𝐴 × 𝐴) ≈ (𝐵𝐶) → (𝐵𝐶) ≈ (𝐴 × 𝐴))
2 bren 8006 . . 3 ((𝐵𝐶) ≈ (𝐴 × 𝐴) ↔ ∃𝑓 𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴))
3 ssdif0 3975 . . . . . 6 (𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ↔ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) = ∅)
4 dmxpid 5377 . . . . . . . . . . . . . 14 dom (𝐴 × 𝐴) = 𝐴
5 f1ofo 6182 . . . . . . . . . . . . . . . . 17 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓:(𝐵𝐶)–onto→(𝐴 × 𝐴))
6 forn 6156 . . . . . . . . . . . . . . . . 17 (𝑓:(𝐵𝐶)–onto→(𝐴 × 𝐴) → ran 𝑓 = (𝐴 × 𝐴))
75, 6syl 17 . . . . . . . . . . . . . . . 16 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → ran 𝑓 = (𝐴 × 𝐴))
8 vex 3234 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
98rnex 7142 . . . . . . . . . . . . . . . 16 ran 𝑓 ∈ V
107, 9syl6eqelr 2739 . . . . . . . . . . . . . . 15 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴 × 𝐴) ∈ V)
11 dmexg 7139 . . . . . . . . . . . . . . 15 ((𝐴 × 𝐴) ∈ V → dom (𝐴 × 𝐴) ∈ V)
1210, 11syl 17 . . . . . . . . . . . . . 14 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → dom (𝐴 × 𝐴) ∈ V)
134, 12syl5eqelr 2735 . . . . . . . . . . . . 13 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝐴 ∈ V)
14 imassrn 5512 . . . . . . . . . . . . . 14 (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ⊆ ran ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓)
15 f1stres 7234 . . . . . . . . . . . . . . . 16 (1st ↾ (𝐴 × 𝐴)):(𝐴 × 𝐴)⟶𝐴
16 f1of 6175 . . . . . . . . . . . . . . . 16 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓:(𝐵𝐶)⟶(𝐴 × 𝐴))
17 fco 6096 . . . . . . . . . . . . . . . 16 (((1st ↾ (𝐴 × 𝐴)):(𝐴 × 𝐴)⟶𝐴𝑓:(𝐵𝐶)⟶(𝐴 × 𝐴)) → ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓):(𝐵𝐶)⟶𝐴)
1815, 16, 17sylancr 696 . . . . . . . . . . . . . . 15 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓):(𝐵𝐶)⟶𝐴)
19 frn 6091 . . . . . . . . . . . . . . 15 (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓):(𝐵𝐶)⟶𝐴 → ran ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) ⊆ 𝐴)
2018, 19syl 17 . . . . . . . . . . . . . 14 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → ran ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) ⊆ 𝐴)
2114, 20syl5ss 3647 . . . . . . . . . . . . 13 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ⊆ 𝐴)
2213, 21ssexd 4838 . . . . . . . . . . . 12 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∈ V)
2322adantr 480 . . . . . . . . . . 11 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∈ V)
24 simpr 476 . . . . . . . . . . 11 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
25 ssdomg 8043 . . . . . . . . . . 11 ((((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∈ V → (𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) → 𝐴 ≼ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)))
2623, 24, 25sylc 65 . . . . . . . . . 10 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝐴 ≼ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
27 domwdom 8520 . . . . . . . . . 10 (𝐴 ≼ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) → 𝐴* (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
2826, 27syl 17 . . . . . . . . 9 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝐴* (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
29 ffun 6086 . . . . . . . . . . . 12 (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓):(𝐵𝐶)⟶𝐴 → Fun ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓))
3018, 29syl 17 . . . . . . . . . . 11 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → Fun ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓))
31 ssun1 3809 . . . . . . . . . . . 12 𝐵 ⊆ (𝐵𝐶)
32 f1odm 6179 . . . . . . . . . . . . 13 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → dom 𝑓 = (𝐵𝐶))
338dmex 7141 . . . . . . . . . . . . 13 dom 𝑓 ∈ V
3432, 33syl6eqelr 2739 . . . . . . . . . . . 12 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐵𝐶) ∈ V)
35 ssexg 4837 . . . . . . . . . . . 12 ((𝐵 ⊆ (𝐵𝐶) ∧ (𝐵𝐶) ∈ V) → 𝐵 ∈ V)
3631, 34, 35sylancr 696 . . . . . . . . . . 11 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝐵 ∈ V)
37 wdomima2g 8532 . . . . . . . . . . 11 ((Fun ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) ∧ 𝐵 ∈ V ∧ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∈ V) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ≼* 𝐵)
3830, 36, 22, 37syl3anc 1366 . . . . . . . . . 10 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ≼* 𝐵)
3938adantr 480 . . . . . . . . 9 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ≼* 𝐵)
40 wdomtr 8521 . . . . . . . . 9 ((𝐴* (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∧ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ≼* 𝐵) → 𝐴* 𝐵)
4128, 39, 40syl2anc 694 . . . . . . . 8 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝐴* 𝐵)
4241orcd 406 . . . . . . 7 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (𝐴* 𝐵𝐴𝐶))
4342ex 449 . . . . . 6 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) → (𝐴* 𝐵𝐴𝐶)))
443, 43syl5bir 233 . . . . 5 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → ((𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) = ∅ → (𝐴* 𝐵𝐴𝐶)))
45 n0 3964 . . . . . 6 ((𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)))
46 ssun2 3810 . . . . . . . . . . . . 13 𝐶 ⊆ (𝐵𝐶)
47 ssexg 4837 . . . . . . . . . . . . 13 ((𝐶 ⊆ (𝐵𝐶) ∧ (𝐵𝐶) ∈ V) → 𝐶 ∈ V)
4846, 34, 47sylancr 696 . . . . . . . . . . . 12 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝐶 ∈ V)
4948adantr 480 . . . . . . . . . . 11 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝐶 ∈ V)
50 f1ofn 6176 . . . . . . . . . . . . . . 15 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓 Fn (𝐵𝐶))
51 elpreima 6377 . . . . . . . . . . . . . . 15 (𝑓 Fn (𝐵𝐶) → (𝑦 ∈ (𝑓 “ ({𝑥} × 𝐴)) ↔ (𝑦 ∈ (𝐵𝐶) ∧ (𝑓𝑦) ∈ ({𝑥} × 𝐴))))
5250, 51syl 17 . . . . . . . . . . . . . 14 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝑦 ∈ (𝑓 “ ({𝑥} × 𝐴)) ↔ (𝑦 ∈ (𝐵𝐶) ∧ (𝑓𝑦) ∈ ({𝑥} × 𝐴))))
5352adantr 480 . . . . . . . . . . . . 13 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑦 ∈ (𝑓 “ ({𝑥} × 𝐴)) ↔ (𝑦 ∈ (𝐵𝐶) ∧ (𝑓𝑦) ∈ ({𝑥} × 𝐴))))
54 elun 3786 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
55 df-or 384 . . . . . . . . . . . . . . . 16 ((𝑦𝐵𝑦𝐶) ↔ (¬ 𝑦𝐵𝑦𝐶))
5654, 55bitri 264 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝐵𝐶) ↔ (¬ 𝑦𝐵𝑦𝐶))
57 eldifn 3766 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → ¬ 𝑥 ∈ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
5857ad2antlr 763 . . . . . . . . . . . . . . . . . 18 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ (𝑓𝑦) ∈ ({𝑥} × 𝐴)) → ¬ 𝑥 ∈ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
5916ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → 𝑓:(𝐵𝐶)⟶(𝐴 × 𝐴))
60 simprr 811 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → 𝑦𝐵)
6131, 60sseldi 3634 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → 𝑦 ∈ (𝐵𝐶))
62 fvco3 6314 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓:(𝐵𝐶)⟶(𝐴 × 𝐴) ∧ 𝑦 ∈ (𝐵𝐶)) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓)‘𝑦) = ((1st ↾ (𝐴 × 𝐴))‘(𝑓𝑦)))
6359, 61, 62syl2anc 694 . . . . . . . . . . . . . . . . . . . . 21 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓)‘𝑦) = ((1st ↾ (𝐴 × 𝐴))‘(𝑓𝑦)))
64 eldifi 3765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝑥𝐴)
6564adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝑥𝐴)
6665snssd 4372 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → {𝑥} ⊆ 𝐴)
67 xpss1 5161 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({𝑥} ⊆ 𝐴 → ({𝑥} × 𝐴) ⊆ (𝐴 × 𝐴))
6866, 67syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ({𝑥} × 𝐴) ⊆ (𝐴 × 𝐴))
6968adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → ({𝑥} × 𝐴) ⊆ (𝐴 × 𝐴))
70 simprl 809 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → (𝑓𝑦) ∈ ({𝑥} × 𝐴))
7169, 70sseldd 3637 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → (𝑓𝑦) ∈ (𝐴 × 𝐴))
72 fvres 6245 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓𝑦) ∈ (𝐴 × 𝐴) → ((1st ↾ (𝐴 × 𝐴))‘(𝑓𝑦)) = (1st ‘(𝑓𝑦)))
7371, 72syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → ((1st ↾ (𝐴 × 𝐴))‘(𝑓𝑦)) = (1st ‘(𝑓𝑦)))
74 xp1st 7242 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓𝑦) ∈ ({𝑥} × 𝐴) → (1st ‘(𝑓𝑦)) ∈ {𝑥})
7570, 74syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → (1st ‘(𝑓𝑦)) ∈ {𝑥})
7673, 75eqeltrd 2730 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → ((1st ↾ (𝐴 × 𝐴))‘(𝑓𝑦)) ∈ {𝑥})
77 elsni 4227 . . . . . . . . . . . . . . . . . . . . . 22 (((1st ↾ (𝐴 × 𝐴))‘(𝑓𝑦)) ∈ {𝑥} → ((1st ↾ (𝐴 × 𝐴))‘(𝑓𝑦)) = 𝑥)
7876, 77syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → ((1st ↾ (𝐴 × 𝐴))‘(𝑓𝑦)) = 𝑥)
7963, 78eqtrd 2685 . . . . . . . . . . . . . . . . . . . 20 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓)‘𝑦) = 𝑥)
80 ffn 6083 . . . . . . . . . . . . . . . . . . . . . . 23 (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓):(𝐵𝐶)⟶𝐴 → ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) Fn (𝐵𝐶))
8118, 80syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) Fn (𝐵𝐶))
8281ad2antrr 762 . . . . . . . . . . . . . . . . . . . . 21 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) Fn (𝐵𝐶))
8331a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → 𝐵 ⊆ (𝐵𝐶))
84 fnfvima 6536 . . . . . . . . . . . . . . . . . . . . 21 ((((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) Fn (𝐵𝐶) ∧ 𝐵 ⊆ (𝐵𝐶) ∧ 𝑦𝐵) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓)‘𝑦) ∈ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
8582, 83, 60, 84syl3anc 1366 . . . . . . . . . . . . . . . . . . . 20 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓)‘𝑦) ∈ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
8679, 85eqeltrrd 2731 . . . . . . . . . . . . . . . . . . 19 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → 𝑥 ∈ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
8786expr 642 . . . . . . . . . . . . . . . . . 18 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ (𝑓𝑦) ∈ ({𝑥} × 𝐴)) → (𝑦𝐵𝑥 ∈ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)))
8858, 87mtod 189 . . . . . . . . . . . . . . . . 17 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ (𝑓𝑦) ∈ ({𝑥} × 𝐴)) → ¬ 𝑦𝐵)
8988ex 449 . . . . . . . . . . . . . . . 16 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ((𝑓𝑦) ∈ ({𝑥} × 𝐴) → ¬ 𝑦𝐵))
9089imim1d 82 . . . . . . . . . . . . . . 15 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ((¬ 𝑦𝐵𝑦𝐶) → ((𝑓𝑦) ∈ ({𝑥} × 𝐴) → 𝑦𝐶)))
9156, 90syl5bi 232 . . . . . . . . . . . . . 14 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑦 ∈ (𝐵𝐶) → ((𝑓𝑦) ∈ ({𝑥} × 𝐴) → 𝑦𝐶)))
9291impd 446 . . . . . . . . . . . . 13 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ((𝑦 ∈ (𝐵𝐶) ∧ (𝑓𝑦) ∈ ({𝑥} × 𝐴)) → 𝑦𝐶))
9353, 92sylbid 230 . . . . . . . . . . . 12 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑦 ∈ (𝑓 “ ({𝑥} × 𝐴)) → 𝑦𝐶))
9493ssrdv 3642 . . . . . . . . . . 11 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑓 “ ({𝑥} × 𝐴)) ⊆ 𝐶)
95 ssdomg 8043 . . . . . . . . . . 11 (𝐶 ∈ V → ((𝑓 “ ({𝑥} × 𝐴)) ⊆ 𝐶 → (𝑓 “ ({𝑥} × 𝐴)) ≼ 𝐶))
9649, 94, 95sylc 65 . . . . . . . . . 10 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑓 “ ({𝑥} × 𝐴)) ≼ 𝐶)
97 f1ocnv 6187 . . . . . . . . . . . . . . 15 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓:(𝐴 × 𝐴)–1-1-onto→(𝐵𝐶))
98 f1of1 6174 . . . . . . . . . . . . . . 15 (𝑓:(𝐴 × 𝐴)–1-1-onto→(𝐵𝐶) → 𝑓:(𝐴 × 𝐴)–1-1→(𝐵𝐶))
9997, 98syl 17 . . . . . . . . . . . . . 14 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓:(𝐴 × 𝐴)–1-1→(𝐵𝐶))
10099adantr 480 . . . . . . . . . . . . 13 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝑓:(𝐴 × 𝐴)–1-1→(𝐵𝐶))
10134adantr 480 . . . . . . . . . . . . 13 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝐵𝐶) ∈ V)
102 snex 4938 . . . . . . . . . . . . . 14 {𝑥} ∈ V
10313adantr 480 . . . . . . . . . . . . . 14 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝐴 ∈ V)
104 xpexg 7002 . . . . . . . . . . . . . 14 (({𝑥} ∈ V ∧ 𝐴 ∈ V) → ({𝑥} × 𝐴) ∈ V)
105102, 103, 104sylancr 696 . . . . . . . . . . . . 13 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ({𝑥} × 𝐴) ∈ V)
106 f1imaen2g 8058 . . . . . . . . . . . . 13 (((𝑓:(𝐴 × 𝐴)–1-1→(𝐵𝐶) ∧ (𝐵𝐶) ∈ V) ∧ (({𝑥} × 𝐴) ⊆ (𝐴 × 𝐴) ∧ ({𝑥} × 𝐴) ∈ V)) → (𝑓 “ ({𝑥} × 𝐴)) ≈ ({𝑥} × 𝐴))
107100, 101, 68, 105, 106syl22anc 1367 . . . . . . . . . . . 12 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑓 “ ({𝑥} × 𝐴)) ≈ ({𝑥} × 𝐴))
108 vex 3234 . . . . . . . . . . . . 13 𝑥 ∈ V
109 xpsnen2g 8094 . . . . . . . . . . . . 13 ((𝑥 ∈ V ∧ 𝐴 ∈ V) → ({𝑥} × 𝐴) ≈ 𝐴)
110108, 103, 109sylancr 696 . . . . . . . . . . . 12 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ({𝑥} × 𝐴) ≈ 𝐴)
111 entr 8049 . . . . . . . . . . . 12 (((𝑓 “ ({𝑥} × 𝐴)) ≈ ({𝑥} × 𝐴) ∧ ({𝑥} × 𝐴) ≈ 𝐴) → (𝑓 “ ({𝑥} × 𝐴)) ≈ 𝐴)
112107, 110, 111syl2anc 694 . . . . . . . . . . 11 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑓 “ ({𝑥} × 𝐴)) ≈ 𝐴)
113 domen1 8143 . . . . . . . . . . 11 ((𝑓 “ ({𝑥} × 𝐴)) ≈ 𝐴 → ((𝑓 “ ({𝑥} × 𝐴)) ≼ 𝐶𝐴𝐶))
114112, 113syl 17 . . . . . . . . . 10 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ((𝑓 “ ({𝑥} × 𝐴)) ≼ 𝐶𝐴𝐶))
11596, 114mpbid 222 . . . . . . . . 9 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝐴𝐶)
116115olcd 407 . . . . . . . 8 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝐴* 𝐵𝐴𝐶))
117116ex 449 . . . . . . 7 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (𝐴* 𝐵𝐴𝐶)))
118117exlimdv 1901 . . . . . 6 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (∃𝑥 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (𝐴* 𝐵𝐴𝐶)))
11945, 118syl5bi 232 . . . . 5 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → ((𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) ≠ ∅ → (𝐴* 𝐵𝐴𝐶)))
12044, 119pm2.61dne 2909 . . . 4 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴* 𝐵𝐴𝐶))
121120exlimiv 1898 . . 3 (∃𝑓 𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴* 𝐵𝐴𝐶))
1222, 121sylbi 207 . 2 ((𝐵𝐶) ≈ (𝐴 × 𝐴) → (𝐴* 𝐵𝐴𝐶))
1231, 122syl 17 1 ((𝐴 × 𝐴) ≈ (𝐵𝐶) → (𝐴* 𝐵𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wex 1744  wcel 2030  wne 2823  Vcvv 3231  cdif 3604  cun 3605  wss 3607  c0 3948  {csn 4210   class class class wbr 4685   × cxp 5141  ccnv 5142  dom cdm 5143  ran crn 5144  cres 5145  cima 5146  ccom 5147  Fun wfun 5920   Fn wfn 5921  wf 5922  1-1wf1 5923  ontowfo 5924  1-1-ontowf1o 5925  cfv 5926  1st c1st 7208  cen 7994  cdom 7995  * cwdom 8503
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-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  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-ral 2946  df-rex 2947  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-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  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-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-1st 7210  df-2nd 7211  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-wdom 8505
This theorem is referenced by:  unxpwdom  8535  ttac  37920
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