Theorem txcn 15467

Description: A map into the product of two topological spaces is continuous iff both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)

Hypotheses

Ref	Expression
txcn.1	⊢ 𝑋 = ∪ 𝑅
txcn.2	⊢ 𝑌 = ∪ 𝑆
txcn.3	⊢ 𝑍 = (𝑋 × 𝑌)
txcn.4	⊢ 𝑊 = ∪ 𝑈
txcn.5	⊢ 𝑃 = (1st ↾ 𝑍)
txcn.6	⊢ 𝑄 = (2nd ↾ 𝑍)

Assertion

Ref	Expression
txcn	⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ↔ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))))

Proof of Theorem txcn

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		txcn.1	. . . . 5 ⊢ 𝑋 = ∪ 𝑅
2	1	toptopon 15210	. . . 4 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
3		txcn.2	. . . . 5 ⊢ 𝑌 = ∪ 𝑆
4	3	toptopon 15210	. . . 4 ⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘𝑌))
5		txcn.5	. . . . . . 7 ⊢ 𝑃 = (1st ↾ 𝑍)
6		txcn.3	. . . . . . . 8 ⊢ 𝑍 = (𝑋 × 𝑌)
7	6	reseq2i 5060	. . . . . . 7 ⊢ (1st ↾ 𝑍) = (1st ↾ (𝑋 × 𝑌))
8	5, 7	eqtri 2259	. . . . . 6 ⊢ 𝑃 = (1st ↾ (𝑋 × 𝑌))
9		tx1cn 15461	. . . . . 6 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
10	8, 9	eqeltrid 2325	. . . . 5 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑃 ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
11		txcn.6	. . . . . . 7 ⊢ 𝑄 = (2nd ↾ 𝑍)
12	6	reseq2i 5060	. . . . . . 7 ⊢ (2nd ↾ 𝑍) = (2nd ↾ (𝑋 × 𝑌))
13	11, 12	eqtri 2259	. . . . . 6 ⊢ 𝑄 = (2nd ↾ (𝑋 × 𝑌))
14		tx2cn 15462	. . . . . 6 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
15	13, 14	eqeltrid 2325	. . . . 5 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑄 ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
16		cnco 15413	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ 𝑃 ∈ ((𝑅 ×t 𝑆) Cn 𝑅)) → (𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅))
17		cnco 15413	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ 𝑄 ∈ ((𝑅 ×t 𝑆) Cn 𝑆)) → (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))
18	16, 17	anim12dan 608	. . . . . 6 ⊢ ((𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ (𝑃 ∈ ((𝑅 ×t 𝑆) Cn 𝑅) ∧ 𝑄 ∈ ((𝑅 ×t 𝑆) Cn 𝑆))) → ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆)))
19	18	expcom 116	. . . . 5 ⊢ ((𝑃 ∈ ((𝑅 ×t 𝑆) Cn 𝑅) ∧ 𝑄 ∈ ((𝑅 ×t 𝑆) Cn 𝑆)) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))))
20	10, 15, 19	syl2anc 415	. . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))))
21	2, 4, 20	syl2anb 291	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))))
22	21	3adant3 1048	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))))
23		cntop1 15393	. . . . . . . 8 ⊢ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) → 𝑈 ∈ Top)
24	23	ad2antrl 494	. . . . . . 7 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) → 𝑈 ∈ Top)
25		txcn.4	. . . . . . . 8 ⊢ 𝑊 = ∪ 𝑈
26	25	topopn 15200	. . . . . . 7 ⊢ (𝑈 ∈ Top → 𝑊 ∈ 𝑈)
27	24, 26	syl 14	. . . . . 6 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) → 𝑊 ∈ 𝑈)
28	25, 1	cnf 15396	. . . . . . 7 ⊢ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) → (𝑃 ∘ 𝐹):𝑊⟶𝑋)
29	28	ad2antrl 494	. . . . . 6 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) → (𝑃 ∘ 𝐹):𝑊⟶𝑋)
30	25, 3	cnf 15396	. . . . . . 7 ⊢ ((𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆) → (𝑄 ∘ 𝐹):𝑊⟶𝑌)
31	30	ad2antll 495	. . . . . 6 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) → (𝑄 ∘ 𝐹):𝑊⟶𝑌)
32	8, 13	upxp 15464	. . . . . . 7 ⊢ ((𝑊 ∈ 𝑈 ∧ (𝑃 ∘ 𝐹):𝑊⟶𝑋 ∧ (𝑄 ∘ 𝐹):𝑊⟶𝑌) → ∃!ℎ(ℎ:𝑊⟶(𝑋 × 𝑌) ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)))
33		feq3 5518	. . . . . . . . . 10 ⊢ (𝑍 = (𝑋 × 𝑌) → (ℎ:𝑊⟶𝑍 ↔ ℎ:𝑊⟶(𝑋 × 𝑌)))
34	6, 33	ax-mp 5	. . . . . . . . 9 ⊢ (ℎ:𝑊⟶𝑍 ↔ ℎ:𝑊⟶(𝑋 × 𝑌))
35	34	3anbi1i 1221	. . . . . . . 8 ⊢ ((ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)) ↔ (ℎ:𝑊⟶(𝑋 × 𝑌) ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)))
36	35	eubii 2095	. . . . . . 7 ⊢ (∃!ℎ(ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)) ↔ ∃!ℎ(ℎ:𝑊⟶(𝑋 × 𝑌) ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)))
37	32, 36	sylibr 134	. . . . . 6 ⊢ ((𝑊 ∈ 𝑈 ∧ (𝑃 ∘ 𝐹):𝑊⟶𝑋 ∧ (𝑄 ∘ 𝐹):𝑊⟶𝑌) → ∃!ℎ(ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)))
38	27, 29, 31, 37	syl3anc 1278	. . . . 5 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) → ∃!ℎ(ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)))
39		euex 2116	. . . . 5 ⊢ (∃!ℎ(ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)) → ∃ℎ(ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)))
40	38, 39	syl 14	. . . 4 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) → ∃ℎ(ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)))
41		simpll3 1069	. . . . . . 7 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))) → 𝐹:𝑊⟶𝑍)
42	27	adantr 276	. . . . . . 7 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))) → 𝑊 ∈ 𝑈)
43	1	topopn 15200	. . . . . . . . . 10 ⊢ (𝑅 ∈ Top → 𝑋 ∈ 𝑅)
44	3	topopn 15200	. . . . . . . . . 10 ⊢ (𝑆 ∈ Top → 𝑌 ∈ 𝑆)
45		xpexg 4889	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑆) → (𝑋 × 𝑌) ∈ V)
46	6, 45	eqeltrid 2325	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑆) → 𝑍 ∈ V)
47	43, 44, 46	syl2an 289	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑍 ∈ V)
48	47	3adant3 1048	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) → 𝑍 ∈ V)
49	48	ad2antrr 492	. . . . . . 7 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))) → 𝑍 ∈ V)
50		fex2 5556	. . . . . . 7 ⊢ ((𝐹:𝑊⟶𝑍 ∧ 𝑊 ∈ 𝑈 ∧ 𝑍 ∈ V) → 𝐹 ∈ V)
51	41, 42, 49, 50	syl3anc 1278	. . . . . 6 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))) → 𝐹 ∈ V)
52		eumo 2118	. . . . . . . 8 ⊢ (∃!ℎ(ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)) → ∃*ℎ(ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)))
53	38, 52	syl 14	. . . . . . 7 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) → ∃*ℎ(ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)))
54	53	adantr 276	. . . . . 6 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))) → ∃*ℎ(ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)))
55		simpr 110	. . . . . 6 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))) → (ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)))
56		3anass 1013	. . . . . . . 8 ⊢ ((ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)) ↔ (ℎ:𝑊⟶𝑍 ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))))
57		coeq2 4938	. . . . . . . . . . . 12 ⊢ (𝐹 = ℎ → (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ))
58		coeq2 4938	. . . . . . . . . . . 12 ⊢ (𝐹 = ℎ → (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))
59	57, 58	jca 306	. . . . . . . . . . 11 ⊢ (𝐹 = ℎ → ((𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)))
60	59	eqcoms 2241	. . . . . . . . . 10 ⊢ (ℎ = 𝐹 → ((𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)))
61	60	biantrud 304	. . . . . . . . 9 ⊢ (ℎ = 𝐹 → (ℎ:𝑊⟶𝑍 ↔ (ℎ:𝑊⟶𝑍 ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)))))
62		feq1 5516	. . . . . . . . 9 ⊢ (ℎ = 𝐹 → (ℎ:𝑊⟶𝑍 ↔ 𝐹:𝑊⟶𝑍))
63	61, 62	bitr3d 190	. . . . . . . 8 ⊢ (ℎ = 𝐹 → ((ℎ:𝑊⟶𝑍 ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))) ↔ 𝐹:𝑊⟶𝑍))
64	56, 63	bitrid 192	. . . . . . 7 ⊢ (ℎ = 𝐹 → ((ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)) ↔ 𝐹:𝑊⟶𝑍))
65	64	moi2 3007	. . . . . 6 ⊢ (((𝐹 ∈ V ∧ ∃*ℎ(ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))) ∧ ((ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)) ∧ 𝐹:𝑊⟶𝑍)) → ℎ = 𝐹)
66	51, 54, 55, 41, 65	syl22anc 1279	. . . . 5 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))) → ℎ = 𝐹)
67		eqid 2238	. . . . . . . . . 10 ⊢ (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
68	67, 1, 3, 6, 5, 11	uptx 15466	. . . . . . . . 9 ⊢ (((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆)) → ∃!ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆))((𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)))
69	68	adantl 277	. . . . . . . 8 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) → ∃!ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆))((𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)))
70		df-reu 2535	. . . . . . . . . 10 ⊢ (∃!ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆))((𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)) ↔ ∃!ℎ(ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))))
71		euex 2116	. . . . . . . . . 10 ⊢ (∃!ℎ(ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))) → ∃ℎ(ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))))
72	70, 71	sylbi 121	. . . . . . . . 9 ⊢ (∃!ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆))((𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)) → ∃ℎ(ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))))
73		eqid 2238	. . . . . . . . . . . . . . 15 ⊢ ∪ (𝑅 ×t 𝑆) = ∪ (𝑅 ×t 𝑆)
74	25, 73	cnf 15396	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → ℎ:𝑊⟶∪ (𝑅 ×t 𝑆))
75	1, 3	txuni 15455	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆))
76	6, 75	eqtrid 2283	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑍 = ∪ (𝑅 ×t 𝑆))
77	76	3adant3 1048	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) → 𝑍 = ∪ (𝑅 ×t 𝑆))
78	77	adantr 276	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) → 𝑍 = ∪ (𝑅 ×t 𝑆))
79	78	feq3d 5522	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) → (ℎ:𝑊⟶𝑍 ↔ ℎ:𝑊⟶∪ (𝑅 ×t 𝑆)))
80	74, 79	imbitrrid 156	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) → (ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → ℎ:𝑊⟶𝑍))
81	80	anim1d 336	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) → ((ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))) → (ℎ:𝑊⟶𝑍 ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)))))
82	81, 56	imbitrrdi 162	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) → ((ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))) → (ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))))
83		simpl 109	. . . . . . . . . . 11 ⊢ ((ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))) → ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆)))
84	82, 83	jca2 308	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) → ((ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))) → ((ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)) ∧ ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆)))))
85	84	eximdv 1933	. . . . . . . . 9 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) → (∃ℎ(ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))) → ∃ℎ((ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)) ∧ ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆)))))
86	72, 85	syl5 32	. . . . . . . 8 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) → (∃!ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆))((𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)) → ∃ℎ((ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)) ∧ ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆)))))
87	69, 86	mpd 13	. . . . . . 7 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) → ∃ℎ((ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)) ∧ ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆))))
88		eupick 2166	. . . . . . 7 ⊢ ((∃!ℎ(ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)) ∧ ∃ℎ((ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)) ∧ ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆)))) → ((ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)) → ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆))))
89	38, 87, 88	syl2anc 415	. . . . . 6 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) → ((ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ)) → ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆))))
90	89	imp 124	. . . . 5 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))) → ℎ ∈ (𝑈 Cn (𝑅 ×t 𝑆)))
91	66, 90	eqeltrrd 2316	. . . 4 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (ℎ:𝑊⟶𝑍 ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ ℎ) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ ℎ))) → 𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)))
92	40, 91	exlimddv 1954	. . 3 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) ∧ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))) → 𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)))
93	92	ex 115	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) → (((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆)) → 𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆))))
94	22, 93	impbid 129	1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ↔ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402  ∃wex 1545  ∃!weu 2086  ∃*wmo 2087   ∈ wcel 2209  ∃!wreu 2530  Vcvv 2821  ∪ cuni 3935   × cxp 4772   ↾ cres 4776   ∘ ccom 4778  ⟶wf 5373  ‘cfv 5377  (class class class)co 6085  1^st c1st 6372  2^nd c2nd 6373  Topctop 15189  TopOnctopon 15202   Cn ccn 15377   ×_t ctx 15444

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-map 6924  df-topgen 13667  df-top 15190  df-topon 15203  df-bases 15235  df-cn 15380  df-tx 15445

This theorem is used by: (None)