Theorem uptx 23541

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 2733	. . . . 5 ⊢ ∪ 𝑈 = ∪ 𝑈
2		eqid 2733	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) = (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
3	1, 2	txcnmpt 23540	. . . 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 7363	. . . 4 ⊢ (𝑈 Cn 𝑇) = (𝑈 Cn (𝑅 ×t 𝑆))
6	3, 5	eleqtrrdi 2844	. . 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 23162	. . . . 5 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝐹:∪ 𝑈⟶𝑋)
9		uptx.3	. . . . . 6 ⊢ 𝑌 = ∪ 𝑆
10	1, 9	cnf 23162	. . . . 5 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝐺:∪ 𝑈⟶𝑌)
11		ffn 6656	. . . . . . . 8 ⊢ (𝐹:∪ 𝑈⟶𝑋 → 𝐹 Fn ∪ 𝑈)
12	11	adantr 480	. . . . . . 7 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → 𝐹 Fn ∪ 𝑈)
13		fo1st 7947	. . . . . . . . . 10 ⊢ 1st :V–onto→V
14		fofn 6742	. . . . . . . . . 10 ⊢ (1st :V–onto→V → 1st Fn V)
15	13, 14	ax-mp 5	. . . . . . . . 9 ⊢ 1st Fn V
16		ssv 3955	. . . . . . . . 9 ⊢ (𝑋 × 𝑌) ⊆ V
17		fnssres 6609	. . . . . . . . 9 ⊢ ((1st Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
18	15, 16, 17	mp2an 692	. . . . . . . 8 ⊢ (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
19		ffvelcdm 7020	. . . . . . . . . . . 12 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝑥 ∈ ∪ 𝑈) → (𝐹‘𝑥) ∈ 𝑋)
20		ffvelcdm 7020	. . . . . . . . . . . 12 ⊢ ((𝐺:∪ 𝑈⟶𝑌 ∧ 𝑥 ∈ ∪ 𝑈) → (𝐺‘𝑥) ∈ 𝑌)
21		opelxpi 5656	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) ∈ 𝑋 ∧ (𝐺‘𝑥) ∈ 𝑌) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝑋 × 𝑌))
22	19, 20, 21	syl2an 596	. . . . . . . . . . 11 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝑥 ∈ ∪ 𝑈) ∧ (𝐺:∪ 𝑈⟶𝑌 ∧ 𝑥 ∈ ∪ 𝑈)) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝑋 × 𝑌))
23	22	anandirs 679	. . . . . . . . . 10 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑥 ∈ ∪ 𝑈) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝑋 × 𝑌))
24	23	fmpttd 7054	. . . . . . . . 9 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩):∪ 𝑈⟶(𝑋 × 𝑌))
25		ffn 6656	. . . . . . . . 9 ⊢ ((𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩):∪ 𝑈⟶(𝑋 × 𝑌) → (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) Fn ∪ 𝑈)
26	24, 25	syl 17	. . . . . . . 8 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) Fn ∪ 𝑈)
27	24	frnd 6664	. . . . . . . 8 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → ran (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ⊆ (𝑋 × 𝑌))
28		fnco 6604	. . . . . . . 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)
29	18, 26, 27, 28	mp3an2i 1468	. . . . . . 7 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → ((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) Fn ∪ 𝑈)
30		fvco3 6927	. . . . . . . . 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)))
31	24, 30	sylan 580	. . . . . . . 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)))
32		fveq2 6828	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
33		fveq2 6828	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑥) = (𝐺‘𝑧))
34	32, 33	opeq12d 4832	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ = ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩)
35		opex 5407	. . . . . . . . . . 11 ⊢ ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩ ∈ V
36	34, 2, 35	fvmpt 6935	. . . . . . . . . 10 ⊢ (𝑧 ∈ ∪ 𝑈 → ((𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑧) = ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩)
37	36	adantl 481	. . . . . . . . 9 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → ((𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑧) = ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩)
38	37	fveq2d 6832	. . . . . . . 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)⟩))
39		ffvelcdm 7020	. . . . . . . . . . . 12 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝑧 ∈ ∪ 𝑈) → (𝐹‘𝑧) ∈ 𝑋)
40		ffvelcdm 7020	. . . . . . . . . . . 12 ⊢ ((𝐺:∪ 𝑈⟶𝑌 ∧ 𝑧 ∈ ∪ 𝑈) → (𝐺‘𝑧) ∈ 𝑌)
41		opelxpi 5656	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑧) ∈ 𝑋 ∧ (𝐺‘𝑧) ∈ 𝑌) → ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩ ∈ (𝑋 × 𝑌))
42	39, 40, 41	syl2an 596	. . . . . . . . . . 11 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝑧 ∈ ∪ 𝑈) ∧ (𝐺:∪ 𝑈⟶𝑌 ∧ 𝑧 ∈ ∪ 𝑈)) → ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩ ∈ (𝑋 × 𝑌))
43	42	anandirs 679	. . . . . . . . . 10 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩ ∈ (𝑋 × 𝑌))
44	43	fvresd 6848	. . . . . . . . 9 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → ((1st ↾ (𝑋 × 𝑌))‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩) = (1st ‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩))
45		fvex 6841	. . . . . . . . . 10 ⊢ (𝐹‘𝑧) ∈ V
46		fvex 6841	. . . . . . . . . 10 ⊢ (𝐺‘𝑧) ∈ V
47	45, 46	op1st 7935	. . . . . . . . 9 ⊢ (1st ‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩) = (𝐹‘𝑧)
48	44, 47	eqtrdi 2784	. . . . . . . 8 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → ((1st ↾ (𝑋 × 𝑌))‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩) = (𝐹‘𝑧))
49	31, 38, 48	3eqtrrd 2773	. . . . . . 7 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → (𝐹‘𝑧) = (((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))‘𝑧))
50	12, 29, 49	eqfnfvd 6973	. . . . . 6 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → 𝐹 = ((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
51		uptx.5	. . . . . . . 8 ⊢ 𝑃 = (1st ↾ 𝑍)
52		uptx.4	. . . . . . . . 9 ⊢ 𝑍 = (𝑋 × 𝑌)
53	52	reseq2i 5929	. . . . . . . 8 ⊢ (1st ↾ 𝑍) = (1st ↾ (𝑋 × 𝑌))
54	51, 53	eqtri 2756	. . . . . . 7 ⊢ 𝑃 = (1st ↾ (𝑋 × 𝑌))
55	54	coeq1i 5803	. . . . . 6 ⊢ (𝑃 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) = ((1st ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))
56	50, 55	eqtr4di 2786	. . . . 5 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → 𝐹 = (𝑃 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
57	8, 10, 56	syl2an 596	. . . 4 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐹 = (𝑃 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
58		ffn 6656	. . . . . . . 8 ⊢ (𝐺:∪ 𝑈⟶𝑌 → 𝐺 Fn ∪ 𝑈)
59	58	adantl 481	. . . . . . 7 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → 𝐺 Fn ∪ 𝑈)
60		fo2nd 7948	. . . . . . . . . 10 ⊢ 2nd :V–onto→V
61		fofn 6742	. . . . . . . . . 10 ⊢ (2nd :V–onto→V → 2nd Fn V)
62	60, 61	ax-mp 5	. . . . . . . . 9 ⊢ 2nd Fn V
63		fnssres 6609	. . . . . . . . 9 ⊢ ((2nd Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
64	62, 16, 63	mp2an 692	. . . . . . . 8 ⊢ (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
65		fnco 6604	. . . . . . . 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)
66	64, 26, 27, 65	mp3an2i 1468	. . . . . . 7 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → ((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) Fn ∪ 𝑈)
67		fvco3 6927	. . . . . . . . 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)))
68	24, 67	sylan 580	. . . . . . . 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)))
69	37	fveq2d 6832	. . . . . . . 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)⟩))
70	43	fvresd 6848	. . . . . . . . 9 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → ((2nd ↾ (𝑋 × 𝑌))‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩) = (2nd ‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩))
71	45, 46	op2nd 7936	. . . . . . . . 9 ⊢ (2nd ‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩) = (𝐺‘𝑧)
72	70, 71	eqtrdi 2784	. . . . . . . 8 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → ((2nd ↾ (𝑋 × 𝑌))‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩) = (𝐺‘𝑧))
73	68, 69, 72	3eqtrrd 2773	. . . . . . 7 ⊢ (((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) ∧ 𝑧 ∈ ∪ 𝑈) → (𝐺‘𝑧) = (((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))‘𝑧))
74	59, 66, 73	eqfnfvd 6973	. . . . . 6 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → 𝐺 = ((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
75		uptx.6	. . . . . . . 8 ⊢ 𝑄 = (2nd ↾ 𝑍)
76	52	reseq2i 5929	. . . . . . . 8 ⊢ (2nd ↾ 𝑍) = (2nd ↾ (𝑋 × 𝑌))
77	75, 76	eqtri 2756	. . . . . . 7 ⊢ 𝑄 = (2nd ↾ (𝑋 × 𝑌))
78	77	coeq1i 5803	. . . . . 6 ⊢ (𝑄 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) = ((2nd ↾ (𝑋 × 𝑌)) ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))
79	74, 78	eqtr4di 2786	. . . . 5 ⊢ ((𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → 𝐺 = (𝑄 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
80	8, 10, 79	syl2an 596	. . . 4 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐺 = (𝑄 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
81	6, 57, 80	jca32 515	. . 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)⟩)))))
82		eleq1 2821	. . . . 5 ⊢ (ℎ = (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) → (ℎ ∈ (𝑈 Cn 𝑇) ↔ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ∈ (𝑈 Cn 𝑇)))
83		coeq2 5802	. . . . . . 7 ⊢ (ℎ = (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) → (𝑃 ∘ ℎ) = (𝑃 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
84	83	eqeq2d 2744	. . . . . 6 ⊢ (ℎ = (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) → (𝐹 = (𝑃 ∘ ℎ) ↔ 𝐹 = (𝑃 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))))
85		coeq2 5802	. . . . . . 7 ⊢ (ℎ = (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) → (𝑄 ∘ ℎ) = (𝑄 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
86	85	eqeq2d 2744	. . . . . 6 ⊢ (ℎ = (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) → (𝐺 = (𝑄 ∘ ℎ) ↔ 𝐺 = (𝑄 ∘ (𝑥 ∈ ∪ 𝑈 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))))
87	84, 86	anbi12d 632	. . . . 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)⟩)))))
88	82, 87	anbi12d 632	. . . 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)⟩))))))
89	88	spcegv 3548	. . 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)))))
90	6, 81, 89	sylc 65	. 2 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃ℎ(ℎ ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))))
91		eqid 2733	. . . . . . . 8 ⊢ ∪ 𝑇 = ∪ 𝑇
92	1, 91	cnf 23162	. . . . . . 7 ⊢ (ℎ ∈ (𝑈 Cn 𝑇) → ℎ:∪ 𝑈⟶∪ 𝑇)
93		cntop2 23157	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑅 ∈ Top)
94		cntop2 23157	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑆 ∈ Top)
95	4	unieqi 4870	. . . . . . . . . 10 ⊢ ∪ 𝑇 = ∪ (𝑅 ×t 𝑆)
96	7, 9	txuni 23508	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆))
97	95, 96	eqtr4id 2787	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ∪ 𝑇 = (𝑋 × 𝑌))
98	93, 94, 97	syl2an 596	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∪ 𝑇 = (𝑋 × 𝑌))
99	98	feq3d 6641	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (ℎ:∪ 𝑈⟶∪ 𝑇 ↔ ℎ:∪ 𝑈⟶(𝑋 × 𝑌)))
100	92, 99	imbitrid 244	. . . . . 6 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (ℎ ∈ (𝑈 Cn 𝑇) → ℎ:∪ 𝑈⟶(𝑋 × 𝑌)))
101	100	anim1d 611	. . . . 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)))))
102		3anass 1094	. . . . 5 ⊢ ((ℎ:∪ 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)) ↔ (ℎ:∪ 𝑈⟶(𝑋 × 𝑌) ∧ (𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))))
103	101, 102	imbitrrdi 252	. . . 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))))
104	103	alrimiv 1928	. . 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))))
105		cntop1 23156	. . . . . 6 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑈 ∈ Top)
106	105	uniexd 7681	. . . . 5 ⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → ∪ 𝑈 ∈ V)
107	54, 77	upxp 23539	. . . . 5 ⊢ ((∪ 𝑈 ∈ V ∧ 𝐹:∪ 𝑈⟶𝑋 ∧ 𝐺:∪ 𝑈⟶𝑌) → ∃!ℎ(ℎ:∪ 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))
108	106, 8, 10, 107	syl2an3an 1424	. . . 4 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃!ℎ(ℎ:∪ 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))
109		eumo 2575	. . . 4 ⊢ (∃!ℎ(ℎ:∪ 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)) → ∃*ℎ(ℎ:∪ 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))
110	108, 109	syl 17	. . 3 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃*ℎ(ℎ:∪ 𝑈⟶(𝑋 × 𝑌) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))
111		moim 2541	. . 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)))))
112	104, 110, 111	sylc 65	. 2 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃*ℎ(ℎ ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))))
113		df-reu 3348	. . 3 ⊢ (∃!ℎ ∈ (𝑈 Cn 𝑇)(𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)) ↔ ∃!ℎ(ℎ ∈ (𝑈 Cn 𝑇) ∧ (𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))))
114		df-eu 2566	. . 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)))))
115	113, 114	bitri 275	. 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)))))
116	90, 112, 115	sylanbrc 583	1 ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃!ℎ ∈ (𝑈 Cn 𝑇)(𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃*wmo 2535  ∃!weu 2565  ∃!wreu 3345  Vcvv 3437   ⊆ wss 3898  ⟨cop 4581  ∪ cuni 4858   ↦ cmpt 5174   × cxp 5617  ran crn 5620   ↾ cres 5621   ∘ ccom 5623   Fn wfn 6481  ⟶wf 6482  –onto→wfo 6484  ‘cfv 6486  (class class class)co 7352  1^st c1st 7925  2^nd c2nd 7926  Topctop 22809   Cn ccn 23140   ×_t ctx 23476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-map 8758  df-topgen 17349  df-top 22810  df-topon 22827  df-bases 22862  df-cn 23143  df-tx 23478

This theorem is referenced by:  txcn  23542