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Theorem unxpwdom2 8376
 Description: Lemma for unxpwdom 8377. (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 7891 . 2 ((𝐴 × 𝐴) ≈ (𝐵𝐶) → (𝐵𝐶) ≈ (𝐴 × 𝐴))
2 bren 7850 . . 3 ((𝐵𝐶) ≈ (𝐴 × 𝐴) ↔ ∃𝑓 𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴))
3 ssdif0 3896 . . . . . 6 (𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ↔ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) = ∅)
4 dmxpid 5266 . . . . . . . . . . . . . 14 dom (𝐴 × 𝐴) = 𝐴
5 f1ofo 6057 . . . . . . . . . . . . . . . . 17 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓:(𝐵𝐶)–onto→(𝐴 × 𝐴))
6 forn 6031 . . . . . . . . . . . . . . . . 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 3176 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
98rnex 6992 . . . . . . . . . . . . . . . 16 ran 𝑓 ∈ V
107, 9syl6eqelr 2697 . . . . . . . . . . . . . . 15 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴 × 𝐴) ∈ V)
11 dmexg 6989 . . . . . . . . . . . . . . 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 2693 . . . . . . . . . . . . 13 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝐴 ∈ V)
14 imassrn 5396 . . . . . . . . . . . . . 14 (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ⊆ ran ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓)
15 f1stres 7081 . . . . . . . . . . . . . . . 16 (1st ↾ (𝐴 × 𝐴)):(𝐴 × 𝐴)⟶𝐴
16 f1of 6050 . . . . . . . . . . . . . . . 16 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓:(𝐵𝐶)⟶(𝐴 × 𝐴))
17 fco 5971 . . . . . . . . . . . . . . . 16 (((1st ↾ (𝐴 × 𝐴)):(𝐴 × 𝐴)⟶𝐴𝑓:(𝐵𝐶)⟶(𝐴 × 𝐴)) → ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓):(𝐵𝐶)⟶𝐴)
1815, 16, 17sylancr 694 . . . . . . . . . . . . . . 15 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓):(𝐵𝐶)⟶𝐴)
19 frn 5966 . . . . . . . . . . . . . . 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 3579 . . . . . . . . . . . . 13 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ⊆ 𝐴)
2213, 21ssexd 4733 . . . . . . . . . . . 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 7887 . . . . . . . . . . 11 ((((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∈ V → (𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) → 𝐴 ≼ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)))
2623, 24, 25sylc 63 . . . . . . . . . 10 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝐴 ≼ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
27 domwdom 8362 . . . . . . . . . 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 5961 . . . . . . . . . . . 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 3738 . . . . . . . . . . . 12 𝐵 ⊆ (𝐵𝐶)
32 f1odm 6054 . . . . . . . . . . . . 13 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → dom 𝑓 = (𝐵𝐶))
338dmex 6991 . . . . . . . . . . . . 13 dom 𝑓 ∈ V
3432, 33syl6eqelr 2697 . . . . . . . . . . . 12 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐵𝐶) ∈ V)
35 ssexg 4732 . . . . . . . . . . . 12 ((𝐵 ⊆ (𝐵𝐶) ∧ (𝐵𝐶) ∈ V) → 𝐵 ∈ V)
3631, 34, 35sylancr 694 . . . . . . . . . . 11 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝐵 ∈ V)
37 wdomima2g 8374 . . . . . . . . . . 11 ((Fun ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) ∧ 𝐵 ∈ V ∧ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∈ V) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ≼* 𝐵)
3830, 36, 22, 37syl3anc 1318 . . . . . . . . . 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 8363 . . . . . . . . 9 ((𝐴* (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∧ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ≼* 𝐵) → 𝐴* 𝐵)
4128, 39, 40syl2anc 691 . . . . . . . 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 232 . . . . 5 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → ((𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) = ∅ → (𝐴* 𝐵𝐴𝐶)))
45 n0 3890 . . . . . 6 ((𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)))
46 ssun2 3739 . . . . . . . . . . . . 13 𝐶 ⊆ (𝐵𝐶)
47 ssexg 4732 . . . . . . . . . . . . 13 ((𝐶 ⊆ (𝐵𝐶) ∧ (𝐵𝐶) ∈ V) → 𝐶 ∈ V)
4846, 34, 47sylancr 694 . . . . . . . . . . . 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 6051 . . . . . . . . . . . . . . 15 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓 Fn (𝐵𝐶))
51 elpreima 6245 . . . . . . . . . . . . . . 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 3715 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
55 df-or 384 . . . . . . . . . . . . . . . 16 ((𝑦𝐵𝑦𝐶) ↔ (¬ 𝑦𝐵𝑦𝐶))
5654, 55bitri 263 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝐵𝐶) ↔ (¬ 𝑦𝐵𝑦𝐶))
57 eldifn 3695 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → ¬ 𝑥 ∈ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
5857ad2antlr 759 . . . . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . . . . . . . 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 792 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → 𝑦𝐵)
6131, 60sseldi 3566 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → 𝑦 ∈ (𝐵𝐶))
62 fvco3 6185 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓:(𝐵𝐶)⟶(𝐴 × 𝐴) ∧ 𝑦 ∈ (𝐵𝐶)) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓)‘𝑦) = ((1st ↾ (𝐴 × 𝐴))‘(𝑓𝑦)))
6359, 61, 62syl2anc 691 . . . . . . . . . . . . . . . . . . . . 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 3694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4281 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → {𝑥} ⊆ 𝐴)
67 xpss1 5151 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 790 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → (𝑓𝑦) ∈ ({𝑥} × 𝐴))
7169, 70sseldd 3569 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → (𝑓𝑦) ∈ (𝐴 × 𝐴))
72 fvres 6117 . . . . . . . . . . . . . . . . . . . . . . . 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 7089 . . . . . . . . . . . . . . . . . . . . . . . 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 2688 . . . . . . . . . . . . . . . . . . . . . 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 4142 . . . . . . . . . . . . . . . . . . . . . 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 2644 . . . . . . . . . . . . . . . . . . . 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 5958 . . . . . . . . . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . . . . . . 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 6400 . . . . . . . . . . . . . . . . . . . . 21 ((((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) Fn (𝐵𝐶) ∧ 𝐵 ⊆ (𝐵𝐶) ∧ 𝑦𝐵) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓)‘𝑦) ∈ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
8582, 83, 60, 84syl3anc 1318 . . . . . . . . . . . . . . . . . . . 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 2689 . . . . . . . . . . . . . . . . . . 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 641 . . . . . . . . . . . . . . . . . 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 188 . . . . . . . . . . . . . . . . 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 80 . . . . . . . . . . . . . . 15 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ((¬ 𝑦𝐵𝑦𝐶) → ((𝑓𝑦) ∈ ({𝑥} × 𝐴) → 𝑦𝐶)))
9156, 90syl5bi 231 . . . . . . . . . . . . . 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 229 . . . . . . . . . . . 12 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑦 ∈ (𝑓 “ ({𝑥} × 𝐴)) → 𝑦𝐶))
9493ssrdv 3574 . . . . . . . . . . 11 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑓 “ ({𝑥} × 𝐴)) ⊆ 𝐶)
95 ssdomg 7887 . . . . . . . . . . 11 (𝐶 ∈ V → ((𝑓 “ ({𝑥} × 𝐴)) ⊆ 𝐶 → (𝑓 “ ({𝑥} × 𝐴)) ≼ 𝐶))
9649, 94, 95sylc 63 . . . . . . . . . 10 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑓 “ ({𝑥} × 𝐴)) ≼ 𝐶)
97 f1ocnv 6062 . . . . . . . . . . . . . . 15 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓:(𝐴 × 𝐴)–1-1-onto→(𝐵𝐶))
98 f1of1 6049 . . . . . . . . . . . . . . 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 4835 . . . . . . . . . . . . . 14 {𝑥} ∈ V
10313adantr 480 . . . . . . . . . . . . . 14 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝐴 ∈ V)
104 xpexg 6858 . . . . . . . . . . . . . 14 (({𝑥} ∈ V ∧ 𝐴 ∈ V) → ({𝑥} × 𝐴) ∈ V)
105102, 103, 104sylancr 694 . . . . . . . . . . . . 13 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ({𝑥} × 𝐴) ∈ V)
106 f1imaen2g 7903 . . . . . . . . . . . . 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 1319 . . . . . . . . . . . 12 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑓 “ ({𝑥} × 𝐴)) ≈ ({𝑥} × 𝐴))
108 vex 3176 . . . . . . . . . . . . 13 𝑥 ∈ V
109 xpsnen2g 7938 . . . . . . . . . . . . 13 ((𝑥 ∈ V ∧ 𝐴 ∈ V) → ({𝑥} × 𝐴) ≈ 𝐴)
110108, 103, 109sylancr 694 . . . . . . . . . . . 12 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ({𝑥} × 𝐴) ≈ 𝐴)
111 entr 7894 . . . . . . . . . . . 12 (((𝑓 “ ({𝑥} × 𝐴)) ≈ ({𝑥} × 𝐴) ∧ ({𝑥} × 𝐴) ≈ 𝐴) → (𝑓 “ ({𝑥} × 𝐴)) ≈ 𝐴)
112107, 110, 111syl2anc 691 . . . . . . . . . . 11 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑓 “ ({𝑥} × 𝐴)) ≈ 𝐴)
113 domen1 7987 . . . . . . . . . . 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 221 . . . . . . . . 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 1848 . . . . . 6 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (∃𝑥 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (𝐴* 𝐵𝐴𝐶)))
11945, 118syl5bi 231 . . . . 5 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → ((𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) ≠ ∅ → (𝐴* 𝐵𝐴𝐶)))
12044, 119pm2.61dne 2868 . . . 4 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴* 𝐵𝐴𝐶))
121120exlimiv 1845 . . 3 (∃𝑓 𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴* 𝐵𝐴𝐶))
1222, 121sylbi 206 . 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 195   ∨ wo 382   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  {csn 4125   class class class wbr 4583   × cxp 5036  ◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041   ∘ ccom 5042  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  1st c1st 7057   ≈ cen 7838   ≼ cdom 7839   ≼* cwdom 8345 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1st 7059  df-2nd 7060  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-wdom 8347 This theorem is referenced by:  unxpwdom  8377  ttac  36621
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