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Theorem unxpwdom2 9036

Description: Lemma for unxpwdom 9037. (Contributed by Mario Carneiro, 15-May-2015.)

**Assertion**
Ref	Expression
unxpwdom2	⊢ ((𝐴 × 𝐴) ≈ (𝐵 ∪ 𝐶) → (𝐴 ≼^* 𝐵 ∨ 𝐴 ≼ 𝐶))

Proof of Theorem unxpwdom2 Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ensym 8541	. 2 ⊢ ((𝐴 × 𝐴) ≈ (𝐵 ∪ 𝐶) → (𝐵 ∪ 𝐶) ≈ (𝐴 × 𝐴))
2		bren 8501	. . 3 ⊢ ((𝐵 ∪ 𝐶) ≈ (𝐴 × 𝐴) ↔ ∃𝑓 𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴))
3		ssdif0 4277	. . . . . 6 ⊢ (𝐴 ⊆ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ↔ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) = ∅)
4		dmxpid 5764	. . . . . . . . . . . . . 14 ⊢ dom (𝐴 × 𝐴) = 𝐴
5		f1ofo 6597	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓:(𝐵 ∪ 𝐶)–onto→(𝐴 × 𝐴))
6		forn 6568	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:(𝐵 ∪ 𝐶)–onto→(𝐴 × 𝐴) → ran 𝑓 = (𝐴 × 𝐴))
7	5, 6	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → ran 𝑓 = (𝐴 × 𝐴))
8		vex 3444	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
9	8	rnex 7599	. . . . . . . . . . . . . . . 16 ⊢ ran 𝑓 ∈ V
10	7, 9	eqeltrrdi 2899	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴 × 𝐴) ∈ V)
11	10	dmexd 7596	. . . . . . . . . . . . . 14 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → dom (𝐴 × 𝐴) ∈ V)
12	4, 11	eqeltrrid 2895	. . . . . . . . . . . . 13 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝐴 ∈ V)
13		imassrn 5907	. . . . . . . . . . . . . 14 ⊢ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ⊆ ran ((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓)
14		f1stres 7695	. . . . . . . . . . . . . . . 16 ⊢ (1^st ↾ (𝐴 × 𝐴)):(𝐴 × 𝐴)⟶𝐴
15		f1of 6590	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓:(𝐵 ∪ 𝐶)⟶(𝐴 × 𝐴))
16		fco 6505	. . . . . . . . . . . . . . . 16 ⊢ (((1^st ↾ (𝐴 × 𝐴)):(𝐴 × 𝐴)⟶𝐴 ∧ 𝑓:(𝐵 ∪ 𝐶)⟶(𝐴 × 𝐴)) → ((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓):(𝐵 ∪ 𝐶)⟶𝐴)
17	14, 15, 16	sylancr 590	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → ((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓):(𝐵 ∪ 𝐶)⟶𝐴)
18	17	frnd 6494	. . . . . . . . . . . . . 14 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → ran ((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) ⊆ 𝐴)
19	13, 18	sstrid 3926	. . . . . . . . . . . . 13 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ⊆ 𝐴)
20	12, 19	ssexd 5192	. . . . . . . . . . . 12 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∈ V)
21	20	adantr 484	. . . . . . . . . . 11 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∈ V)
22		simpr 488	. . . . . . . . . . 11 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝐴 ⊆ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
23		ssdomg 8538	. . . . . . . . . . 11 ⊢ ((((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∈ V → (𝐴 ⊆ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) → 𝐴 ≼ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)))
24	21, 22, 23	sylc 65	. . . . . . . . . 10 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝐴 ≼ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
25		domwdom 9022	. . . . . . . . . 10 ⊢ (𝐴 ≼ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) → 𝐴 ≼^* (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
26	24, 25	syl 17	. . . . . . . . 9 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝐴 ≼^* (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
27	17	ffund 6491	. . . . . . . . . . 11 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → Fun ((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓))
28		ssun1 4099	. . . . . . . . . . . 12 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶)
29		f1odm 6594	. . . . . . . . . . . . 13 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → dom 𝑓 = (𝐵 ∪ 𝐶))
30	8	dmex 7598	. . . . . . . . . . . . 13 ⊢ dom 𝑓 ∈ V
31	29, 30	eqeltrrdi 2899	. . . . . . . . . . . 12 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐵 ∪ 𝐶) ∈ V)
32		ssexg 5191	. . . . . . . . . . . 12 ⊢ ((𝐵 ⊆ (𝐵 ∪ 𝐶) ∧ (𝐵 ∪ 𝐶) ∈ V) → 𝐵 ∈ V)
33	28, 31, 32	sylancr 590	. . . . . . . . . . 11 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝐵 ∈ V)
34		wdomima2g 9034	. . . . . . . . . . 11 ⊢ ((Fun ((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) ∧ 𝐵 ∈ V ∧ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∈ V) → (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ≼^* 𝐵)
35	27, 33, 20, 34	syl3anc 1368	. . . . . . . . . 10 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ≼^* 𝐵)
36	35	adantr 484	. . . . . . . . 9 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ≼^* 𝐵)
37		wdomtr 9023	. . . . . . . . 9 ⊢ ((𝐴 ≼^* (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∧ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ≼^* 𝐵) → 𝐴 ≼^* 𝐵)
38	26, 36, 37	syl2anc 587	. . . . . . . 8 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝐴 ≼^* 𝐵)
39	38	orcd 870	. . . . . . 7 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (𝐴 ≼^* 𝐵 ∨ 𝐴 ≼ 𝐶))
40	39	ex 416	. . . . . 6 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴 ⊆ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) → (𝐴 ≼^* 𝐵 ∨ 𝐴 ≼ 𝐶)))
41	3, 40	syl5bir 246	. . . . 5 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → ((𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) = ∅ → (𝐴 ≼^* 𝐵 ∨ 𝐴 ≼ 𝐶)))
42		n0 4260	. . . . . 6 ⊢ ((𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)))
43		ssun2 4100	. . . . . . . . . . . . 13 ⊢ 𝐶 ⊆ (𝐵 ∪ 𝐶)
44		ssexg 5191	. . . . . . . . . . . . 13 ⊢ ((𝐶 ⊆ (𝐵 ∪ 𝐶) ∧ (𝐵 ∪ 𝐶) ∈ V) → 𝐶 ∈ V)
45	43, 31, 44	sylancr 590	. . . . . . . . . . . 12 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝐶 ∈ V)
46	45	adantr 484	. . . . . . . . . . 11 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝐶 ∈ V)
47		f1ofn 6591	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓 Fn (𝐵 ∪ 𝐶))
48		elpreima 6805	. . . . . . . . . . . . . . 15 ⊢ (𝑓 Fn (𝐵 ∪ 𝐶) → (𝑦 ∈ (^◡𝑓 “ ({𝑥} × 𝐴)) ↔ (𝑦 ∈ (𝐵 ∪ 𝐶) ∧ (𝑓‘𝑦) ∈ ({𝑥} × 𝐴))))
49	47, 48	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝑦 ∈ (^◡𝑓 “ ({𝑥} × 𝐴)) ↔ (𝑦 ∈ (𝐵 ∪ 𝐶) ∧ (𝑓‘𝑦) ∈ ({𝑥} × 𝐴))))
50	49	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑦 ∈ (^◡𝑓 “ ({𝑥} × 𝐴)) ↔ (𝑦 ∈ (𝐵 ∪ 𝐶) ∧ (𝑓‘𝑦) ∈ ({𝑥} × 𝐴))))
51		elun 4076	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶))
52		df-or 845	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶) ↔ (¬ 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
53	51, 52	bitri 278	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (¬ 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
54		eldifn 4055	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → ¬ 𝑥 ∈ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
55	54	ad2antlr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ (𝑓‘𝑦) ∈ ({𝑥} × 𝐴)) → ¬ 𝑥 ∈ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
56	15	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → 𝑓:(𝐵 ∪ 𝐶)⟶(𝐴 × 𝐴))
57		simprr 772	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
58	28, 57	sseldi 3913	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ (𝐵 ∪ 𝐶))
59		fvco3 6737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓:(𝐵 ∪ 𝐶)⟶(𝐴 × 𝐴) ∧ 𝑦 ∈ (𝐵 ∪ 𝐶)) → (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓)‘𝑦) = ((1^st ↾ (𝐴 × 𝐴))‘(𝑓‘𝑦)))
60	56, 58, 59	syl2anc 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓)‘𝑦) = ((1^st ↾ (𝐴 × 𝐴))‘(𝑓‘𝑦)))
61		eldifi 4054	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝑥 ∈ 𝐴)
62	61	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝑥 ∈ 𝐴)
63	62	snssd 4702	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → {𝑥} ⊆ 𝐴)
64		xpss1 5538	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ({𝑥} ⊆ 𝐴 → ({𝑥} × 𝐴) ⊆ (𝐴 × 𝐴))
65	63, 64	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ({𝑥} × 𝐴) ⊆ (𝐴 × 𝐴))
66	65	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → ({𝑥} × 𝐴) ⊆ (𝐴 × 𝐴))
67		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → (𝑓‘𝑦) ∈ ({𝑥} × 𝐴))
68	66, 67	sseldd 3916	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → (𝑓‘𝑦) ∈ (𝐴 × 𝐴))
69	68	fvresd 6665	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → ((1^st ↾ (𝐴 × 𝐴))‘(𝑓‘𝑦)) = (1^st ‘(𝑓‘𝑦)))
70		xp1st 7703	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) → (1^st ‘(𝑓‘𝑦)) ∈ {𝑥})
71	67, 70	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → (1^st ‘(𝑓‘𝑦)) ∈ {𝑥})
72	69, 71	eqeltrd 2890	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → ((1^st ↾ (𝐴 × 𝐴))‘(𝑓‘𝑦)) ∈ {𝑥})
73		elsni 4542	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1^st ↾ (𝐴 × 𝐴))‘(𝑓‘𝑦)) ∈ {𝑥} → ((1^st ↾ (𝐴 × 𝐴))‘(𝑓‘𝑦)) = 𝑥)
74	72, 73	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → ((1^st ↾ (𝐴 × 𝐴))‘(𝑓‘𝑦)) = 𝑥)
75	60, 74	eqtrd 2833	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓)‘𝑦) = 𝑥)
76	17	ffnd 6488	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → ((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) Fn (𝐵 ∪ 𝐶))
77	76	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → ((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) Fn (𝐵 ∪ 𝐶))
78	28	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → 𝐵 ⊆ (𝐵 ∪ 𝐶))
79		fnfvima 6973	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) Fn (𝐵 ∪ 𝐶) ∧ 𝐵 ⊆ (𝐵 ∪ 𝐶) ∧ 𝑦 ∈ 𝐵) → (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓)‘𝑦) ∈ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
80	77, 78, 57, 79	syl3anc 1368	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓)‘𝑦) ∈ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
81	75, 80	eqeltrrd 2891	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
82	81	expr 460	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ (𝑓‘𝑦) ∈ ({𝑥} × 𝐴)) → (𝑦 ∈ 𝐵 → 𝑥 ∈ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)))
83	55, 82	mtod 201	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ (𝑓‘𝑦) ∈ ({𝑥} × 𝐴)) → ¬ 𝑦 ∈ 𝐵)
84	83	ex 416	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) → ¬ 𝑦 ∈ 𝐵))
85	84	imim1d 82	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ((¬ 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) → ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) → 𝑦 ∈ 𝐶)))
86	53, 85	syl5bi 245	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑦 ∈ (𝐵 ∪ 𝐶) → ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) → 𝑦 ∈ 𝐶)))
87	86	impd 414	. . . . . . . . . . . . 13 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ((𝑦 ∈ (𝐵 ∪ 𝐶) ∧ (𝑓‘𝑦) ∈ ({𝑥} × 𝐴)) → 𝑦 ∈ 𝐶))
88	50, 87	sylbid 243	. . . . . . . . . . . 12 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑦 ∈ (^◡𝑓 “ ({𝑥} × 𝐴)) → 𝑦 ∈ 𝐶))
89	88	ssrdv 3921	. . . . . . . . . . 11 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (^◡𝑓 “ ({𝑥} × 𝐴)) ⊆ 𝐶)
90		ssdomg 8538	. . . . . . . . . . 11 ⊢ (𝐶 ∈ V → ((^◡𝑓 “ ({𝑥} × 𝐴)) ⊆ 𝐶 → (^◡𝑓 “ ({𝑥} × 𝐴)) ≼ 𝐶))
91	46, 89, 90	sylc 65	. . . . . . . . . 10 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (^◡𝑓 “ ({𝑥} × 𝐴)) ≼ 𝐶)
92		f1ocnv 6602	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → ^◡𝑓:(𝐴 × 𝐴)–1-1-onto→(𝐵 ∪ 𝐶))
93		f1of1 6589	. . . . . . . . . . . . . . 15 ⊢ (^◡𝑓:(𝐴 × 𝐴)–1-1-onto→(𝐵 ∪ 𝐶) → ^◡𝑓:(𝐴 × 𝐴)–1-1→(𝐵 ∪ 𝐶))
94	92, 93	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → ^◡𝑓:(𝐴 × 𝐴)–1-1→(𝐵 ∪ 𝐶))
95	94	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ^◡𝑓:(𝐴 × 𝐴)–1-1→(𝐵 ∪ 𝐶))
96	31	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝐵 ∪ 𝐶) ∈ V)
97		snex 5297	. . . . . . . . . . . . . 14 ⊢ {𝑥} ∈ V
98	12	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝐴 ∈ V)
99		xpexg 7453	. . . . . . . . . . . . . 14 ⊢ (({𝑥} ∈ V ∧ 𝐴 ∈ V) → ({𝑥} × 𝐴) ∈ V)
100	97, 98, 99	sylancr 590	. . . . . . . . . . . . 13 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ({𝑥} × 𝐴) ∈ V)
101		f1imaen2g 8553	. . . . . . . . . . . . 13 ⊢ (((^◡𝑓:(𝐴 × 𝐴)–1-1→(𝐵 ∪ 𝐶) ∧ (𝐵 ∪ 𝐶) ∈ V) ∧ (({𝑥} × 𝐴) ⊆ (𝐴 × 𝐴) ∧ ({𝑥} × 𝐴) ∈ V)) → (^◡𝑓 “ ({𝑥} × 𝐴)) ≈ ({𝑥} × 𝐴))
102	95, 96, 65, 100, 101	syl22anc 837	. . . . . . . . . . . 12 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (^◡𝑓 “ ({𝑥} × 𝐴)) ≈ ({𝑥} × 𝐴))
103		vex 3444	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
104		xpsnen2g 8593	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ V) → ({𝑥} × 𝐴) ≈ 𝐴)
105	103, 98, 104	sylancr 590	. . . . . . . . . . . 12 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ({𝑥} × 𝐴) ≈ 𝐴)
106		entr 8544	. . . . . . . . . . . 12 ⊢ (((^◡𝑓 “ ({𝑥} × 𝐴)) ≈ ({𝑥} × 𝐴) ∧ ({𝑥} × 𝐴) ≈ 𝐴) → (^◡𝑓 “ ({𝑥} × 𝐴)) ≈ 𝐴)
107	102, 105, 106	syl2anc 587	. . . . . . . . . . 11 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (^◡𝑓 “ ({𝑥} × 𝐴)) ≈ 𝐴)
108		domen1 8643	. . . . . . . . . . 11 ⊢ ((^◡𝑓 “ ({𝑥} × 𝐴)) ≈ 𝐴 → ((^◡𝑓 “ ({𝑥} × 𝐴)) ≼ 𝐶 ↔ 𝐴 ≼ 𝐶))
109	107, 108	syl 17	. . . . . . . . . 10 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ((^◡𝑓 “ ({𝑥} × 𝐴)) ≼ 𝐶 ↔ 𝐴 ≼ 𝐶))
110	91, 109	mpbid 235	. . . . . . . . 9 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝐴 ≼ 𝐶)
111	110	olcd 871	. . . . . . . 8 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝐴 ≼^* 𝐵 ∨ 𝐴 ≼ 𝐶))
112	111	ex 416	. . . . . . 7 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (𝐴 ≼^* 𝐵 ∨ 𝐴 ≼ 𝐶)))
113	112	exlimdv 1934	. . . . . 6 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (∃𝑥 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (𝐴 ≼^* 𝐵 ∨ 𝐴 ≼ 𝐶)))
114	42, 113	syl5bi 245	. . . . 5 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → ((𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) ≠ ∅ → (𝐴 ≼^* 𝐵 ∨ 𝐴 ≼ 𝐶)))
115	41, 114	pm2.61dne 3073	. . . 4 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴 ≼^* 𝐵 ∨ 𝐴 ≼ 𝐶))
116	115	exlimiv 1931	. . 3 ⊢ (∃𝑓 𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴 ≼^* 𝐵 ∨ 𝐴 ≼ 𝐶))
117	2, 116	sylbi 220	. 2 ⊢ ((𝐵 ∪ 𝐶) ≈ (𝐴 × 𝐴) → (𝐴 ≼^* 𝐵 ∨ 𝐴 ≼ 𝐶))
118	1, 117	syl 17	1 ⊢ ((𝐴 × 𝐴) ≈ (𝐵 ∪ 𝐶) → (𝐴 ≼^* 𝐵 ∨ 𝐴 ≼ 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∖ cdif 3878 ∪ cun 3879 ⊆ wss 3881 ∅c0 4243 {csn 4525 class class class wbr 5030 × cxp 5517 ^◡ccnv 5518 dom cdm 5519 ran crn 5520 ↾ cres 5521 “ cima 5522 ∘ ccom 5523 Fun wfun 6318 Fn wfn 6319 ⟶wf 6320 –1-1→wf1 6321 –onto→wfo 6322 –1-1-onto→wf1o 6323 ‘cfv 6324 1^st c1st 7669 ≈ cen 8489 ≼ cdom 8490 ≼^* cwdom 9012

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-1st 7671 df-2nd 7672 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-wdom 9013

This theorem is referenced by: unxpwdom 9037 ttac 39977

W3C validator