Theorem uptx 15065

Description: Universal property of the binary topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)

Hypotheses

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

Assertion

Ref	Expression
uptx	⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃!ℎ ∈ (𝑈 Cn 𝑇)(𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))

Distinct variable groups:   ℎ,𝐹   ℎ,𝐺   𝑃,ℎ   𝑄,ℎ   𝑅,ℎ   𝑇,ℎ   𝑆,ℎ   𝑈,ℎ   ℎ,𝑋   ℎ,𝑌

Allowed substitution hint:   𝑍(ℎ)

Proof of Theorem uptx

Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2231	. . . . 5 ⊢ ∪ 𝑈 = ∪ 𝑈
2		eqid 2231	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) = (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
3	1, 2	txcnmpt 15064	. . . 4 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ∈ (𝑈 Cn (𝑅 ×t 𝑆)))
4		uptx.1	. . . . 5 ⊢ 𝑇 = (𝑅 ×t 𝑆)
5	4	oveq2i 6039	. . . 4 ⊢ (𝑈 Cn 𝑇) = (𝑈 Cn (𝑅 ×t 𝑆))
6	3, 5	eleqtrrdi 2325	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ∈ (𝑈 Cn 𝑇))
7		uptx.2	. . . . . 6 ⊢ 𝑋 = ∪ 𝑅
8	1, 7	cnf 14995	. . . . 5 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝐹:∪ 𝑈⟶𝑋)
9		uptx.3	. . . . . 6 ⊢ 𝑌 = ∪ 𝑆
10	1, 9	cnf 14995	. . . . 5 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝐺:∪ 𝑈⟶𝑌)
11		ffn 5489	. . . . . . . 8 ⊢ (𝐹:∪ 𝑈⟶𝑋 → 𝐹 Fn ∪ 𝑈)
12	11	adantr 276	. . . . . . 7 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → 𝐹 Fn ∪ 𝑈)
13		fo1st 6329	. . . . . . . . . 10 ⊢ 1st :V–onto→V
14		fofn 5570	. . . . . . . . . 10 ⊢ (1st :V–onto→V → 1st Fn V)
15	13, 14	ax-mp 5	. . . . . . . . 9 ⊢ 1st Fn V
16		ssv 3250	. . . . . . . . 9 ⊢ (𝑋 × 𝑌) ⊆ V
17		fnssres 5452	. . . . . . . . 9 ⊢ ((1st Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
18	15, 16, 17	mp2an 426	. . . . . . . 8 ⊢ (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
19		ffvelcdm 5788	. . . . . . . . . . . 12 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝑥 ∈ ∪ 𝑈) → (𝐹‘𝑥) ∈ 𝑋)
20		ffvelcdm 5788	. . . . . . . . . . . 12 ⊢ ((𝐺:∪ 𝑈⟶𝑌 ∧ 𝑥 ∈ ∪ 𝑈) → (𝐺‘𝑥) ∈ 𝑌)
21		opelxpi 4763	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) ∈ 𝑋 ∧ (𝐺‘𝑥) ∈ 𝑌) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝑋 × 𝑌))
22	19, 20, 21	syl2an 289	. . . . . . . . . . 11 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝑥 ∈ ∪ 𝑈) ∧ (𝐺:∪ 𝑈⟶𝑌 ∧ 𝑥 ∈ ∪ 𝑈)) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝑋 × 𝑌))
23	22	anandirs 597	. . . . . . . . . 10 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑥 ∈ ∪ 𝑈) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝑋 × 𝑌))
24	23	fmpttd 5810	. . . . . . . . 9 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩):∪ 𝑈⟶(𝑋 × 𝑌))
25	24	ffnd 5490	. . . . . . . 8 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) Fn ∪ 𝑈)
26	24	frnd 5499	. . . . . . . 8 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → ran (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ⊆ (𝑋 × 𝑌))
27		fnco 5447	. . . . . . . 8 ⊢ (((1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) ∧ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) Fn ∪ 𝑈 ∧ ran (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ⊆ (𝑋 × 𝑌)) → ((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) Fn ∪ 𝑈)
28	18, 25, 26, 27	mp3an2i 1379	. . . . . . 7 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → ((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) Fn ∪ 𝑈)
29		fvco3 5726	. . . . . . . . 9 ⊢ (((𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩):∪ 𝑈⟶(𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → (((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))‘𝑧) = ((1st ↾ (𝑋 × 𝑌))‘((𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑧)))
30	24, 29	sylan 283	. . . . . . . 8 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → (((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))‘𝑧) = ((1st ↾ (𝑋 × 𝑌))‘((𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑧)))
31		fveq2 5648	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
32		fveq2 5648	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑥) = (𝐺‘𝑧))
33	31, 32	opeq12d 3875	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ = ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩)
34		simpr 110	. . . . . . . . . 10 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → 𝑧 ∈ ∪ 𝑈)
35		simpll 527	. . . . . . . . . . . 12 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → 𝐹:∪ 𝑈⟶𝑋)
36	35, 34	ffvelcdmd 5791	. . . . . . . . . . 11 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → (𝐹‘𝑧) ∈ 𝑋)
37		simplr 529	. . . . . . . . . . . 12 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → 𝐺:∪ 𝑈⟶𝑌)
38	37, 34	ffvelcdmd 5791	. . . . . . . . . . 11 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → (𝐺‘𝑧) ∈ 𝑌)
39	36, 38	opelxpd 4764	. . . . . . . . . 10 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩ ∈ (𝑋 × 𝑌))
40	2, 33, 34, 39	fvmptd3 5749	. . . . . . . . 9 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → ((𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑧) = ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩)
41	40	fveq2d 5652	. . . . . . . 8 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → ((1st ↾ (𝑋 × 𝑌))‘((𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑧)) = ((1st ↾ (𝑋 × 𝑌))‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩))
42		ffvelcdm 5788	. . . . . . . . . . . 12 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝑧 ∈ ∪ 𝑈) → (𝐹‘𝑧) ∈ 𝑋)
43		ffvelcdm 5788	. . . . . . . . . . . 12 ⊢ ((𝐺:∪ 𝑈⟶𝑌 ∧ 𝑧 ∈ ∪ 𝑈) → (𝐺‘𝑧) ∈ 𝑌)
44		opelxpi 4763	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑧) ∈ 𝑋 ∧ (𝐺‘𝑧) ∈ 𝑌) → ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩ ∈ (𝑋 × 𝑌))
45	42, 43, 44	syl2an 289	. . . . . . . . . . 11 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝑧 ∈ ∪ 𝑈) ∧ (𝐺:∪ 𝑈⟶𝑌 ∧ 𝑧 ∈ ∪ 𝑈)) → ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩ ∈ (𝑋 × 𝑌))
46	45	anandirs 597	. . . . . . . . . 10 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩ ∈ (𝑋 × 𝑌))
47	46	fvresd 5673	. . . . . . . . 9 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → ((1st ↾ (𝑋 × 𝑌))‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩) = (1st ‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩))
48		op1stg 6322	. . . . . . . . . 10 ⊢ (((𝐹‘𝑧) ∈ 𝑋 ∧ (𝐺‘𝑧) ∈ 𝑌) → (1st ‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩) = (𝐹‘𝑧))
49	36, 38, 48	syl2anc 411	. . . . . . . . 9 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → (1st ‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩) = (𝐹‘𝑧))
50	47, 49	eqtrd 2264	. . . . . . . 8 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → ((1st ↾ (𝑋 × 𝑌))‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩) = (𝐹‘𝑧))
51	30, 41, 50	3eqtrrd 2269	. . . . . . 7 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → (𝐹‘𝑧) = (((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))‘𝑧))
52	12, 28, 51	eqfnfvd 5756	. . . . . 6 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → 𝐹 = ((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
53		uptx.5	. . . . . . . 8 ⊢ 𝑃 = (1st ↾ 𝑍)
54		uptx.4	. . . . . . . . 9 ⊢ 𝑍 = (𝑋 × 𝑌)
55	54	reseq2i 5016	. . . . . . . 8 ⊢ (1st ↾ 𝑍) = (1st ↾ (𝑋 × 𝑌))
56	53, 55	eqtri 2252	. . . . . . 7 ⊢ 𝑃 = (1st ↾ (𝑋 × 𝑌))
57	56	coeq1i 4895	. . . . . 6 ⊢ (𝑃 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) = ((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))
58	52, 57	eqtr4di 2282	. . . . 5 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → 𝐹 = (𝑃 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
59	8, 10, 58	syl2an 289	. . . 4 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐹 = (𝑃 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
60		ffn 5489	. . . . . . . 8 ⊢ (𝐺:∪ 𝑈⟶𝑌 → 𝐺 Fn ∪ 𝑈)
61	60	adantl 277	. . . . . . 7 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → 𝐺 Fn ∪ 𝑈)
62		fo2nd 6330	. . . . . . . . . 10 ⊢ 2nd :V–onto→V
63		fofn 5570	. . . . . . . . . 10 ⊢ (2nd :V–onto→V → 2nd Fn V)
64	62, 63	ax-mp 5	. . . . . . . . 9 ⊢ 2nd Fn V
65		fnssres 5452	. . . . . . . . 9 ⊢ ((2nd Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
66	64, 16, 65	mp2an 426	. . . . . . . 8 ⊢ (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
67		fnco 5447	. . . . . . . 8 ⊢ (((2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) ∧ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) Fn ∪ 𝑈 ∧ ran (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ⊆ (𝑋 × 𝑌)) → ((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) Fn ∪ 𝑈)
68	66, 25, 26, 67	mp3an2i 1379	. . . . . . 7 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → ((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) Fn ∪ 𝑈)
69		fvco3 5726	. . . . . . . . 9 ⊢ (((𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩):∪ 𝑈⟶(𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → (((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))‘𝑧) = ((2nd ↾ (𝑋 × 𝑌))‘((𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑧)))
70	24, 69	sylan 283	. . . . . . . 8 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → (((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))‘𝑧) = ((2nd ↾ (𝑋 × 𝑌))‘((𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑧)))
71	40	fveq2d 5652	. . . . . . . 8 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → ((2nd ↾ (𝑋 × 𝑌))‘((𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑧)) = ((2nd ↾ (𝑋 × 𝑌))‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩))
72	46	fvresd 5673	. . . . . . . . 9 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → ((2nd ↾ (𝑋 × 𝑌))‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩) = (2nd ‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩))
73		op2ndg 6323	. . . . . . . . . 10 ⊢ (((𝐹‘𝑧) ∈ 𝑋 ∧ (𝐺‘𝑧) ∈ 𝑌) → (2nd ‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩) = (𝐺‘𝑧))
74	36, 38, 73	syl2anc 411	. . . . . . . . 9 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → (2nd ‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩) = (𝐺‘𝑧))
75	72, 74	eqtrd 2264	. . . . . . . 8 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → ((2nd ↾ (𝑋 × 𝑌))‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩) = (𝐺‘𝑧))
76	70, 71, 75	3eqtrrd 2269	. . . . . . 7 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → (𝐺‘𝑧) = (((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))‘𝑧))
77	61, 68, 76	eqfnfvd 5756	. . . . . 6 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → 𝐺 = ((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
78		uptx.6	. . . . . . . 8 ⊢ 𝑄 = (2nd ↾ 𝑍)
79	54	reseq2i 5016	. . . . . . . 8 ⊢ (2nd ↾ 𝑍) = (2nd ↾ (𝑋 × 𝑌))
80	78, 79	eqtri 2252	. . . . . . 7 ⊢ 𝑄 = (2nd ↾ (𝑋 × 𝑌))
81	80	coeq1i 4895	. . . . . 6 ⊢ (𝑄 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) = ((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))
82	77, 81	eqtr4di 2282	. . . . 5 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → 𝐺 = (𝑄 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
83	8, 10, 82	syl2an 289	. . . 4 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐺 = (𝑄 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
84	6, 59, 83	jca32 310	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ((𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) ∧ 𝐺 = (𝑄 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))))
85		eleq1 2294	. . . . 5 ⊢ (ℎ = (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) → (ℎ ∈ (𝑈 Cn 𝑇) ↔ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ∈ (𝑈 Cn 𝑇)))
86		coeq2 4894	. . . . . . 7 ⊢ (ℎ = (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) → (𝑃 ∘ ℎ) = (𝑃 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
87	86	eqeq2d 2243	. . . . . 6 ⊢ (ℎ = (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) → (𝐹 = (𝑃 ∘ ℎ) ↔ 𝐹 = (𝑃 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))))
88		coeq2 4894	. . . . . . 7 ⊢ (ℎ = (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) → (𝑄 ∘ ℎ) = (𝑄 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
89	88	eqeq2d 2243	. . . . . 6 ⊢ (ℎ = (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) → (𝐺 = (𝑄 ∘ ℎ) ↔ 𝐺 = (𝑄 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))))
90	87, 89	anbi12d 473	. . . . 5 ⊢ (ℎ = (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) → ((𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)) ↔ (𝐹 = (𝑃 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) ∧ 𝐺 = (𝑄 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))))
91	85, 90	anbi12d 473	. . . 4 ⊢ (ℎ = (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) → ((ℎ ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) ↔ ((𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) ∧ 𝐺 = (𝑄 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))))))
92	91	spcegv 2895	. . 3 ⊢ ((𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ∈ (𝑈 Cn 𝑇) → (((𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) ∧ 𝐺 = (𝑄 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))) → ∃ℎ(ℎ ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))))
93	6, 84, 92	sylc 62	. 2 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃ℎ(ℎ ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))))
94		eqid 2231	. . . . . . . 8 ⊢ ∪ 𝑇 = ∪ 𝑇
95	1, 94	cnf 14995	. . . . . . 7 ⊢ (ℎ ∈ (𝑈 Cn 𝑇) → ℎ:∪ 𝑈⟶∪ 𝑇)
96		cntop2 14993	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑅 ∈ Top)
97		cntop2 14993	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑆 ∈ Top)
98	4	unieqi 3908	. . . . . . . . . 10 ⊢ ∪ 𝑇 = ∪ (𝑅 ×t 𝑆)
99	7, 9	txuni 15054	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆))
100	98, 99	eqtr4id 2283	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∪ 𝑇 = (𝑋 × 𝑌))
101	96, 97, 100	syl2an 289	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∪ 𝑇 = (𝑋 × 𝑌))
102	101	feq3d 5478	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (ℎ:∪ 𝑈⟶∪ 𝑇 ↔ ℎ:∪ 𝑈⟶(𝑋 × 𝑌)))
103	95, 102	imbitrid 154	. . . . . 6 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (ℎ ∈ (𝑈 Cn 𝑇) → ℎ:∪ 𝑈⟶(𝑋 × 𝑌)))
104	103	anim1d 336	. . . . 5 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ((ℎ ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) → (ℎ:∪ 𝑈⟶(𝑋 × 𝑌) ∧ (𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))))
105		3anass 1009	. . . . 5 ⊢ ((ℎ:∪ 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)) ↔ (ℎ:∪ 𝑈⟶(𝑋 × 𝑌) ∧ (𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))))
106	104, 105	imbitrrdi 162	. . . 4 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ((ℎ ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) → (ℎ:∪ 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))))
107	106	alrimiv 1922	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∀ℎ((ℎ ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) → (ℎ:∪ 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))))
108		cntop1 14992	. . . . . 6 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑈 ∈ Top)
109		uniexg 4542	. . . . . 6 ⊢ (𝑈 ∈ Top → ∪ 𝑈 ∈ V)
110	108, 109	syl 14	. . . . 5 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → ∪ 𝑈 ∈ V)
111	56, 80	upxp 15063	. . . . 5 ⊢ ((∪ 𝑈 ∈ V ∧ 𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → ∃!ℎ(ℎ:∪ 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))
112	110, 8, 10, 111	syl2an3an 1335	. . . 4 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃!ℎ(ℎ:∪ 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))
113		eumo 2111	. . . 4 ⊢ (∃!ℎ(ℎ:∪ 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)) → ∃*ℎ(ℎ:∪ 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))
114	112, 113	syl 14	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃*ℎ(ℎ:∪ 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))
115		moim 2144	. . 3 ⊢ (∀ℎ((ℎ ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) → (ℎ:∪ 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) → (∃*ℎ(ℎ:∪ 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)) → ∃*ℎ(ℎ ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))))
116	107, 114, 115	sylc 62	. 2 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃*ℎ(ℎ ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))))
117		df-reu 2518	. . 3 ⊢ (∃!ℎ ∈ (𝑈 Cn 𝑇)(𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)) ↔ ∃!ℎ(ℎ ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))))
118		eu5 2127	. . 3 ⊢ (∃!ℎ(ℎ ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) ↔ (∃ℎ(ℎ ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) ∧ ∃*ℎ(ℎ ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))))
119	117, 118	bitri 184	. 2 ⊢ (∃!ℎ ∈ (𝑈 Cn 𝑇)(𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)) ↔ (∃ℎ(ℎ ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) ∧ ∃*ℎ(ℎ ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))))
120	93, 116, 119	sylanbrc 417	1 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃!ℎ ∈ (𝑈 Cn 𝑇)(𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005  ∀wal 1396   = wceq 1398  ∃wex 1541  ∃!weu 2079  ∃*wmo 2080   ∈ wcel 2202  ∃!wreu 2513  Vcvv 2803   ⊆ wss 3201  ⟨cop 3676  ∪ cuni 3898   ↦ cmpt 4155   × cxp 4729  ran crn 4732   ↾ cres 4733   ∘ ccom 4735   Fn wfn 5328  ⟶wf 5329  –onto→wfo 5331  ‘cfv 5333  (class class class)co 6028  1^st c1st 6310  2^nd c2nd 6311  Topctop 14788   Cn ccn 14976   ×_t ctx 15043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-map 6862  df-topgen 13404  df-top 14789  df-topon 14802  df-bases 14834  df-cn 14979  df-tx 15044

This theorem is referenced by:  txcn  15066