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Theorem unxpwdom2 8762
 Description: Lemma for unxpwdom 8763. (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 8271 . 2 ((𝐴 × 𝐴) ≈ (𝐵𝐶) → (𝐵𝐶) ≈ (𝐴 × 𝐴))
2 bren 8231 . . 3 ((𝐵𝐶) ≈ (𝐴 × 𝐴) ↔ ∃𝑓 𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴))
3 ssdif0 4171 . . . . . 6 (𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ↔ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) = ∅)
4 dmxpid 5577 . . . . . . . . . . . . . 14 dom (𝐴 × 𝐴) = 𝐴
5 f1ofo 6385 . . . . . . . . . . . . . . . . 17 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓:(𝐵𝐶)–onto→(𝐴 × 𝐴))
6 forn 6356 . . . . . . . . . . . . . . . . 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 3417 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
98rnex 7362 . . . . . . . . . . . . . . . 16 ran 𝑓 ∈ V
107, 9syl6eqelr 2915 . . . . . . . . . . . . . . 15 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴 × 𝐴) ∈ V)
1110dmexd 7360 . . . . . . . . . . . . . 14 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → dom (𝐴 × 𝐴) ∈ V)
124, 11syl5eqelr 2911 . . . . . . . . . . . . 13 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝐴 ∈ V)
13 imassrn 5718 . . . . . . . . . . . . . 14 (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ⊆ ran ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓)
14 f1stres 7452 . . . . . . . . . . . . . . . 16 (1st ↾ (𝐴 × 𝐴)):(𝐴 × 𝐴)⟶𝐴
15 f1of 6378 . . . . . . . . . . . . . . . 16 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓:(𝐵𝐶)⟶(𝐴 × 𝐴))
16 fco 6295 . . . . . . . . . . . . . . . 16 (((1st ↾ (𝐴 × 𝐴)):(𝐴 × 𝐴)⟶𝐴𝑓:(𝐵𝐶)⟶(𝐴 × 𝐴)) → ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓):(𝐵𝐶)⟶𝐴)
1714, 15, 16sylancr 583 . . . . . . . . . . . . . . 15 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓):(𝐵𝐶)⟶𝐴)
1817frnd 6285 . . . . . . . . . . . . . 14 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → ran ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) ⊆ 𝐴)
1913, 18syl5ss 3838 . . . . . . . . . . . . 13 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ⊆ 𝐴)
2012, 19ssexd 5030 . . . . . . . . . . . 12 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∈ V)
2120adantr 474 . . . . . . . . . . 11 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∈ V)
22 simpr 479 . . . . . . . . . . 11 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
23 ssdomg 8268 . . . . . . . . . . 11 ((((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∈ V → (𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) → 𝐴 ≼ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)))
2421, 22, 23sylc 65 . . . . . . . . . 10 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝐴 ≼ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
25 domwdom 8748 . . . . . . . . . 10 (𝐴 ≼ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) → 𝐴* (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
2624, 25syl 17 . . . . . . . . 9 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝐴* (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
2717ffund 6282 . . . . . . . . . . 11 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → Fun ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓))
28 ssun1 4003 . . . . . . . . . . . 12 𝐵 ⊆ (𝐵𝐶)
29 f1odm 6382 . . . . . . . . . . . . 13 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → dom 𝑓 = (𝐵𝐶))
308dmex 7361 . . . . . . . . . . . . 13 dom 𝑓 ∈ V
3129, 30syl6eqelr 2915 . . . . . . . . . . . 12 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐵𝐶) ∈ V)
32 ssexg 5029 . . . . . . . . . . . 12 ((𝐵 ⊆ (𝐵𝐶) ∧ (𝐵𝐶) ∈ V) → 𝐵 ∈ V)
3328, 31, 32sylancr 583 . . . . . . . . . . 11 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝐵 ∈ V)
34 wdomima2g 8760 . . . . . . . . . . 11 ((Fun ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) ∧ 𝐵 ∈ V ∧ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∈ V) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ≼* 𝐵)
3527, 33, 20, 34syl3anc 1496 . . . . . . . . . 10 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ≼* 𝐵)
3635adantr 474 . . . . . . . . 9 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ≼* 𝐵)
37 wdomtr 8749 . . . . . . . . 9 ((𝐴* (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∧ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ≼* 𝐵) → 𝐴* 𝐵)
3826, 36, 37syl2anc 581 . . . . . . . 8 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝐴* 𝐵)
3938orcd 906 . . . . . . 7 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (𝐴* 𝐵𝐴𝐶))
4039ex 403 . . . . . 6 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) → (𝐴* 𝐵𝐴𝐶)))
413, 40syl5bir 235 . . . . 5 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → ((𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) = ∅ → (𝐴* 𝐵𝐴𝐶)))
42 n0 4160 . . . . . 6 ((𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)))
43 ssun2 4004 . . . . . . . . . . . . 13 𝐶 ⊆ (𝐵𝐶)
44 ssexg 5029 . . . . . . . . . . . . 13 ((𝐶 ⊆ (𝐵𝐶) ∧ (𝐵𝐶) ∈ V) → 𝐶 ∈ V)
4543, 31, 44sylancr 583 . . . . . . . . . . . 12 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝐶 ∈ V)
4645adantr 474 . . . . . . . . . . 11 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝐶 ∈ V)
47 f1ofn 6379 . . . . . . . . . . . . . . 15 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓 Fn (𝐵𝐶))
48 elpreima 6586 . . . . . . . . . . . . . . 15 (𝑓 Fn (𝐵𝐶) → (𝑦 ∈ (𝑓 “ ({𝑥} × 𝐴)) ↔ (𝑦 ∈ (𝐵𝐶) ∧ (𝑓𝑦) ∈ ({𝑥} × 𝐴))))
4947, 48syl 17 . . . . . . . . . . . . . 14 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝑦 ∈ (𝑓 “ ({𝑥} × 𝐴)) ↔ (𝑦 ∈ (𝐵𝐶) ∧ (𝑓𝑦) ∈ ({𝑥} × 𝐴))))
5049adantr 474 . . . . . . . . . . . . 13 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑦 ∈ (𝑓 “ ({𝑥} × 𝐴)) ↔ (𝑦 ∈ (𝐵𝐶) ∧ (𝑓𝑦) ∈ ({𝑥} × 𝐴))))
51 elun 3980 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
52 df-or 881 . . . . . . . . . . . . . . . 16 ((𝑦𝐵𝑦𝐶) ↔ (¬ 𝑦𝐵𝑦𝐶))
5351, 52bitri 267 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝐵𝐶) ↔ (¬ 𝑦𝐵𝑦𝐶))
54 eldifn 3960 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → ¬ 𝑥 ∈ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
5554ad2antlr 720 . . . . . . . . . . . . . . . . . 18 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ (𝑓𝑦) ∈ ({𝑥} × 𝐴)) → ¬ 𝑥 ∈ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
5615ad2antrr 719 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → 𝑓:(𝐵𝐶)⟶(𝐴 × 𝐴))
57 simprr 791 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → 𝑦𝐵)
5828, 57sseldi 3825 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → 𝑦 ∈ (𝐵𝐶))
59 fvco3 6522 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓:(𝐵𝐶)⟶(𝐴 × 𝐴) ∧ 𝑦 ∈ (𝐵𝐶)) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓)‘𝑦) = ((1st ↾ (𝐴 × 𝐴))‘(𝑓𝑦)))
6056, 58, 59syl2anc 581 . . . . . . . . . . . . . . . . . . . . 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)))
61 eldifi 3959 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝑥𝐴)
6261adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝑥𝐴)
6362snssd 4558 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → {𝑥} ⊆ 𝐴)
64 xpss1 5361 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({𝑥} ⊆ 𝐴 → ({𝑥} × 𝐴) ⊆ (𝐴 × 𝐴))
6563, 64syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ({𝑥} × 𝐴) ⊆ (𝐴 × 𝐴))
6665adantr 474 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → ({𝑥} × 𝐴) ⊆ (𝐴 × 𝐴))
67 simprl 789 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → (𝑓𝑦) ∈ ({𝑥} × 𝐴))
6866, 67sseldd 3828 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → (𝑓𝑦) ∈ (𝐴 × 𝐴))
69 fvres 6452 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓𝑦) ∈ (𝐴 × 𝐴) → ((1st ↾ (𝐴 × 𝐴))‘(𝑓𝑦)) = (1st ‘(𝑓𝑦)))
7068, 69syl 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)))
71 xp1st 7460 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓𝑦) ∈ ({𝑥} × 𝐴) → (1st ‘(𝑓𝑦)) ∈ {𝑥})
7267, 71syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → (1st ‘(𝑓𝑦)) ∈ {𝑥})
7370, 72eqeltrd 2906 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → ((1st ↾ (𝐴 × 𝐴))‘(𝑓𝑦)) ∈ {𝑥})
74 elsni 4414 . . . . . . . . . . . . . . . . . . . . . 22 (((1st ↾ (𝐴 × 𝐴))‘(𝑓𝑦)) ∈ {𝑥} → ((1st ↾ (𝐴 × 𝐴))‘(𝑓𝑦)) = 𝑥)
7573, 74syl 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)
7660, 75eqtrd 2861 . . . . . . . . . . . . . . . . . . . 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)
7717ffnd 6279 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) Fn (𝐵𝐶))
7877ad2antrr 719 . . . . . . . . . . . . . . . . . . . . 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))
7928a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦𝐵)) → 𝐵 ⊆ (𝐵𝐶))
80 fnfvima 6752 . . . . . . . . . . . . . . . . . . . . 21 ((((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) Fn (𝐵𝐶) ∧ 𝐵 ⊆ (𝐵𝐶) ∧ 𝑦𝐵) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓)‘𝑦) ∈ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
8178, 79, 57, 80syl3anc 1496 . . . . . . . . . . . . . . . . . . . 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))
8276, 81eqeltrrd 2907 . . . . . . . . . . . . . . . . . . 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))
8382expr 450 . . . . . . . . . . . . . . . . . 18 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ (𝑓𝑦) ∈ ({𝑥} × 𝐴)) → (𝑦𝐵𝑥 ∈ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)))
8455, 83mtod 190 . . . . . . . . . . . . . . . . 17 (((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ (𝑓𝑦) ∈ ({𝑥} × 𝐴)) → ¬ 𝑦𝐵)
8584ex 403 . . . . . . . . . . . . . . . 16 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ((𝑓𝑦) ∈ ({𝑥} × 𝐴) → ¬ 𝑦𝐵))
8685imim1d 82 . . . . . . . . . . . . . . 15 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ((¬ 𝑦𝐵𝑦𝐶) → ((𝑓𝑦) ∈ ({𝑥} × 𝐴) → 𝑦𝐶)))
8753, 86syl5bi 234 . . . . . . . . . . . . . 14 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑦 ∈ (𝐵𝐶) → ((𝑓𝑦) ∈ ({𝑥} × 𝐴) → 𝑦𝐶)))
8887impd 400 . . . . . . . . . . . . 13 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ((𝑦 ∈ (𝐵𝐶) ∧ (𝑓𝑦) ∈ ({𝑥} × 𝐴)) → 𝑦𝐶))
8950, 88sylbid 232 . . . . . . . . . . . 12 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑦 ∈ (𝑓 “ ({𝑥} × 𝐴)) → 𝑦𝐶))
9089ssrdv 3833 . . . . . . . . . . 11 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑓 “ ({𝑥} × 𝐴)) ⊆ 𝐶)
91 ssdomg 8268 . . . . . . . . . . 11 (𝐶 ∈ V → ((𝑓 “ ({𝑥} × 𝐴)) ⊆ 𝐶 → (𝑓 “ ({𝑥} × 𝐴)) ≼ 𝐶))
9246, 90, 91sylc 65 . . . . . . . . . 10 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑓 “ ({𝑥} × 𝐴)) ≼ 𝐶)
93 f1ocnv 6390 . . . . . . . . . . . . . . 15 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓:(𝐴 × 𝐴)–1-1-onto→(𝐵𝐶))
94 f1of1 6377 . . . . . . . . . . . . . . 15 (𝑓:(𝐴 × 𝐴)–1-1-onto→(𝐵𝐶) → 𝑓:(𝐴 × 𝐴)–1-1→(𝐵𝐶))
9593, 94syl 17 . . . . . . . . . . . . . 14 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓:(𝐴 × 𝐴)–1-1→(𝐵𝐶))
9695adantr 474 . . . . . . . . . . . . 13 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝑓:(𝐴 × 𝐴)–1-1→(𝐵𝐶))
9731adantr 474 . . . . . . . . . . . . 13 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝐵𝐶) ∈ V)
98 snex 5129 . . . . . . . . . . . . . 14 {𝑥} ∈ V
9912adantr 474 . . . . . . . . . . . . . 14 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝐴 ∈ V)
100 xpexg 7220 . . . . . . . . . . . . . 14 (({𝑥} ∈ V ∧ 𝐴 ∈ V) → ({𝑥} × 𝐴) ∈ V)
10198, 99, 100sylancr 583 . . . . . . . . . . . . 13 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ({𝑥} × 𝐴) ∈ V)
102 f1imaen2g 8283 . . . . . . . . . . . . 13 (((𝑓:(𝐴 × 𝐴)–1-1→(𝐵𝐶) ∧ (𝐵𝐶) ∈ V) ∧ (({𝑥} × 𝐴) ⊆ (𝐴 × 𝐴) ∧ ({𝑥} × 𝐴) ∈ V)) → (𝑓 “ ({𝑥} × 𝐴)) ≈ ({𝑥} × 𝐴))
10396, 97, 65, 101, 102syl22anc 874 . . . . . . . . . . . 12 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑓 “ ({𝑥} × 𝐴)) ≈ ({𝑥} × 𝐴))
104 vex 3417 . . . . . . . . . . . . 13 𝑥 ∈ V
105 xpsnen2g 8322 . . . . . . . . . . . . 13 ((𝑥 ∈ V ∧ 𝐴 ∈ V) → ({𝑥} × 𝐴) ≈ 𝐴)
106104, 99, 105sylancr 583 . . . . . . . . . . . 12 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ({𝑥} × 𝐴) ≈ 𝐴)
107 entr 8274 . . . . . . . . . . . 12 (((𝑓 “ ({𝑥} × 𝐴)) ≈ ({𝑥} × 𝐴) ∧ ({𝑥} × 𝐴) ≈ 𝐴) → (𝑓 “ ({𝑥} × 𝐴)) ≈ 𝐴)
108103, 106, 107syl2anc 581 . . . . . . . . . . 11 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑓 “ ({𝑥} × 𝐴)) ≈ 𝐴)
109 domen1 8371 . . . . . . . . . . 11 ((𝑓 “ ({𝑥} × 𝐴)) ≈ 𝐴 → ((𝑓 “ ({𝑥} × 𝐴)) ≼ 𝐶𝐴𝐶))
110108, 109syl 17 . . . . . . . . . 10 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ((𝑓 “ ({𝑥} × 𝐴)) ≼ 𝐶𝐴𝐶))
11192, 110mpbid 224 . . . . . . . . 9 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝐴𝐶)
112111olcd 907 . . . . . . . 8 ((𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝐴* 𝐵𝐴𝐶))
113112ex 403 . . . . . . 7 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (𝐴* 𝐵𝐴𝐶)))
114113exlimdv 2034 . . . . . 6 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (∃𝑥 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (𝐴* 𝐵𝐴𝐶)))
11542, 114syl5bi 234 . . . . 5 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → ((𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) ≠ ∅ → (𝐴* 𝐵𝐴𝐶)))
11641, 115pm2.61dne 3085 . . . 4 (𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴* 𝐵𝐴𝐶))
117116exlimiv 2031 . . 3 (∃𝑓 𝑓:(𝐵𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴* 𝐵𝐴𝐶))
1182, 117sylbi 209 . 2 ((𝐵𝐶) ≈ (𝐴 × 𝐴) → (𝐴* 𝐵𝐴𝐶))
1191, 118syl 17 1 ((𝐴 × 𝐴) ≈ (𝐵𝐶) → (𝐴* 𝐵𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 880   = wceq 1658  ∃wex 1880   ∈ wcel 2166   ≠ wne 2999  Vcvv 3414   ∖ cdif 3795   ∪ cun 3796   ⊆ wss 3798  ∅c0 4144  {csn 4397   class class class wbr 4873   × cxp 5340  ◡ccnv 5341  dom cdm 5342  ran crn 5343   ↾ cres 5344   “ cima 5345   ∘ ccom 5346  Fun wfun 6117   Fn wfn 6118  ⟶wf 6119  –1-1→wf1 6120  –onto→wfo 6121  –1-1-onto→wf1o 6122  ‘cfv 6123  1st c1st 7426   ≈ cen 8219   ≼ cdom 8220   ≼* cwdom 8731 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-1st 7428  df-2nd 7429  df-er 8009  df-en 8223  df-dom 8224  df-sdom 8225  df-wdom 8733 This theorem is referenced by:  unxpwdom  8763  ttac  38446
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