unxpwdom2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > unxpwdom2		Structured version Visualization version GIF version

Theorem unxpwdom2 9524

Description: Lemma for unxpwdom 9525. (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 8943	. 2 ⊢ ((𝐴 × 𝐴) ≈ (𝐵 ∪ 𝐶) → (𝐵 ∪ 𝐶) ≈ (𝐴 × 𝐴))
2		bren 8893	. . 3 ⊢ ((𝐵 ∪ 𝐶) ≈ (𝐴 × 𝐴) ↔ ∃𝑓 𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴))
3		ssdif0 4323	. . . . . 6 ⊢ (𝐴 ⊆ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ↔ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) = ∅)
4		dmxpid 5885	. . . . . . . . . . . . . 14 ⊢ dom (𝐴 × 𝐴) = 𝐴
5		f1ofo 6791	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓:(𝐵 ∪ 𝐶)–onto→(𝐴 × 𝐴))
6		forn 6759	. . . . . . . . . . . . . . . . 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 3449	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
9	8	rnex 7849	. . . . . . . . . . . . . . . 16 ⊢ ran 𝑓 ∈ V
10	7, 9	eqeltrrdi 2847	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴 × 𝐴) ∈ V)
11	10	dmexd 7842	. . . . . . . . . . . . . 14 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → dom (𝐴 × 𝐴) ∈ V)
12	4, 11	eqeltrrid 2843	. . . . . . . . . . . . 13 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝐴 ∈ V)
13		imassrn 6024	. . . . . . . . . . . . . 14 ⊢ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ⊆ ran ((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓)
14		f1stres 7945	. . . . . . . . . . . . . . . 16 ⊢ (1^st ↾ (𝐴 × 𝐴)):(𝐴 × 𝐴)⟶𝐴
15		f1of 6784	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓:(𝐵 ∪ 𝐶)⟶(𝐴 × 𝐴))
16		fco 6692	. . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → ((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓):(𝐵 ∪ 𝐶)⟶𝐴)
18	17	frnd 6676	. . . . . . . . . . . . . 14 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → ran ((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) ⊆ 𝐴)
19	13, 18	sstrid 3955	. . . . . . . . . . . . 13 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ⊆ 𝐴)
20	12, 19	ssexd 5281	. . . . . . . . . . . 12 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∈ V)
21	20	adantr 481	. . . . . . . . . . 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 485	. . . . . . . . . . 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 8940	. . . . . . . . . . 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 9510	. . . . . . . . . 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 6672	. . . . . . . . . . 11 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → Fun ((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓))
28		ssun1 4132	. . . . . . . . . . . 12 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶)
29		f1odm 6788	. . . . . . . . . . . . 13 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → dom 𝑓 = (𝐵 ∪ 𝐶))
30	8	dmex 7848	. . . . . . . . . . . . 13 ⊢ dom 𝑓 ∈ V
31	29, 30	eqeltrrdi 2847	. . . . . . . . . . . 12 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐵 ∪ 𝐶) ∈ V)
32		ssexg 5280	. . . . . . . . . . . 12 ⊢ ((𝐵 ⊆ (𝐵 ∪ 𝐶) ∧ (𝐵 ∪ 𝐶) ∈ V) → 𝐵 ∈ V)
33	28, 31, 32	sylancr 587	. . . . . . . . . . 11 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝐵 ∈ V)
34		wdomima2g 9522	. . . . . . . . . . 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 1371	. . . . . . . . . 10 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ≼^* 𝐵)
36	35	adantr 481	. . . . . . . . 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 9511	. . . . . . . . 9 ⊢ ((𝐴 ≼^* (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∧ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ≼^* 𝐵) → 𝐴 ≼^* 𝐵)
38	26, 36, 37	syl2anc 584	. . . . . . . 8 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝐴 ≼^* 𝐵)
39	38	orcd 871	. . . . . . 7 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (𝐴 ≼^* 𝐵 ∨ 𝐴 ≼ 𝐶))
40	39	ex 413	. . . . . 6 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴 ⊆ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) → (𝐴 ≼^* 𝐵 ∨ 𝐴 ≼ 𝐶)))
41	3, 40	biimtrrid 242	. . . . 5 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → ((𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) = ∅ → (𝐴 ≼^* 𝐵 ∨ 𝐴 ≼ 𝐶)))
42		n0 4306	. . . . . 6 ⊢ ((𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)))
43		ssun2 4133	. . . . . . . . . . . . 13 ⊢ 𝐶 ⊆ (𝐵 ∪ 𝐶)
44		ssexg 5280	. . . . . . . . . . . . 13 ⊢ ((𝐶 ⊆ (𝐵 ∪ 𝐶) ∧ (𝐵 ∪ 𝐶) ∈ V) → 𝐶 ∈ V)
45	43, 31, 44	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝐶 ∈ V)
46	45	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝐶 ∈ V)
47		f1ofn 6785	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓 Fn (𝐵 ∪ 𝐶))
48		elpreima 7008	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 4108	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶))
52		df-or 846	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶) ↔ (¬ 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
53	51, 52	bitri 274	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (¬ 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
54		eldifn 4087	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → ¬ 𝑥 ∈ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))
55	54	ad2antlr 725	. . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . . . . . . . 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	sselid 3942	. . . . . . . . . . . . . . . . . . . . . 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 6940	. . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . 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 4086	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝑥 ∈ 𝐴)
62	61	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝑥 ∈ 𝐴)
63	62	snssd 4769	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → {𝑥} ⊆ 𝐴)
64		xpss1 5652	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . . . . . . 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 3945	. . . . . . . . . . . . . . . . . . . . . . . 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 6862	. . . . . . . . . . . . . . . . . . . . . . 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 7953	. . . . . . . . . . . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . . . . . . . . . . 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 4603	. . . . . . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . . . . . 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 6669	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → ((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) Fn (𝐵 ∪ 𝐶))
77	76	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 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 7183	. . . . . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . . . . 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 457	. . . . . . . . . . . . . . . . . 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 197	. . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . 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	biimtrid 241	. . . . . . . . . . . . . 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 411	. . . . . . . . . . . . 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 239	. . . . . . . . . . . 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 3950	. . . . . . . . . . 11 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (^◡𝑓 “ ({𝑥} × 𝐴)) ⊆ 𝐶)
90		ssdomg 8940	. . . . . . . . . . 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 6796	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → ^◡𝑓:(𝐴 × 𝐴)–1-1-onto→(𝐵 ∪ 𝐶))
93		f1of1 6783	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝐵 ∪ 𝐶) ∈ V)
97		vsnex 5386	. . . . . . . . . . . . . 14 ⊢ {𝑥} ∈ V
98	12	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝐴 ∈ V)
99		xpexg 7684	. . . . . . . . . . . . . 14 ⊢ (({𝑥} ∈ V ∧ 𝐴 ∈ V) → ({𝑥} × 𝐴) ∈ V)
100	97, 98, 99	sylancr 587	. . . . . . . . . . . . 13 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ({𝑥} × 𝐴) ∈ V)
101		f1imaen2g 8955	. . . . . . . . . . . . 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 3449	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
104		xpsnen2g 9009	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ V) → ({𝑥} × 𝐴) ≈ 𝐴)
105	103, 98, 104	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ({𝑥} × 𝐴) ≈ 𝐴)
106		entr 8946	. . . . . . . . . . . 12 ⊢ (((^◡𝑓 “ ({𝑥} × 𝐴)) ≈ ({𝑥} × 𝐴) ∧ ({𝑥} × 𝐴) ≈ 𝐴) → (^◡𝑓 “ ({𝑥} × 𝐴)) ≈ 𝐴)
107	102, 105, 106	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (^◡𝑓 “ ({𝑥} × 𝐴)) ≈ 𝐴)
108		domen1 9063	. . . . . . . . . . 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 231	. . . . . . . . 9 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝐴 ≼ 𝐶)
111	110	olcd 872	. . . . . . . 8 ⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝐴 ≼^* 𝐵 ∨ 𝐴 ≼ 𝐶))
112	111	ex 413	. . . . . . 7 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝑥 ∈ (𝐴 ∖ (((1^st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (𝐴 ≼^* 𝐵 ∨ 𝐴 ≼ 𝐶)))
113	112	exlimdv 1936	. . . . . 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	biimtrid 241	. . . . 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 3031	. . . 4 ⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴 ≼^* 𝐵 ∨ 𝐴 ≼ 𝐶))
116	115	exlimiv 1933	. . 3 ⊢ (∃𝑓 𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴 ≼^* 𝐵 ∨ 𝐴 ≼ 𝐶))
117	2, 116	sylbi 216	. 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 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2943 Vcvv 3445 ∖ cdif 3907 ∪ cun 3908 ⊆ wss 3910 ∅c0 4282 {csn 4586 class class class wbr 5105 × cxp 5631 ^◡ccnv 5632 dom cdm 5633 ran crn 5634 ↾ cres 5635 “ cima 5636 ∘ ccom 5637 Fun wfun 6490 Fn wfn 6491 ⟶wf 6492 –1-1→wf1 6493 –onto→wfo 6494 –1-1-onto→wf1o 6495 ‘cfv 6496 1^st c1st 7919 ≈ cen 8880 ≼ cdom 8881 ≼^* cwdom 9500

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-1st 7921 df-2nd 7922 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-wdom 9501

This theorem is referenced by: unxpwdom 9525 ttac 41346

W3C validator