Theorem upxp 23598

Description: Universal property of the Cartesian product considered as a categorical product in the category of sets. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)

Hypotheses

Ref	Expression
upxp.1	⊢ 𝑃 = (1st ↾ (𝐵 × 𝐶))
upxp.2	⊢ 𝑄 = (2nd ↾ (𝐵 × 𝐶))

Assertion

Ref	Expression
upxp	⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → ∃!ℎ(ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))

Distinct variable groups:   𝐴,ℎ   𝐵,ℎ   𝐶,ℎ   ℎ,𝐹   ℎ,𝐺   𝐷,ℎ

Allowed substitution hints:   𝑃(ℎ)   𝑄(ℎ)

Proof of Theorem upxp

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

Step	Hyp	Ref	Expression
1		mptexg 7169	. . . 4 ⊢ (𝐴 ∈ 𝐷 → (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ∈ V)
2		eueq 3655	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ∈ V ↔ ∃!ℎ ℎ = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))
3	1, 2	sylib 218	. . 3 ⊢ (𝐴 ∈ 𝐷 → ∃!ℎ ℎ = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))
4	3	3ad2ant1 1134	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → ∃!ℎ ℎ = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))
5		ffn 6662	. . . . . . . 8 ⊢ (ℎ:𝐴⟶(𝐵 × 𝐶) → ℎ Fn 𝐴)
6	5	3ad2ant1 1134	. . . . . . 7 ⊢ ((ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)) → ℎ Fn 𝐴)
7	6	adantl 481	. . . . . 6 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ (ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) → ℎ Fn 𝐴)
8		ffvelcdm 7027	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
9		ffvelcdm 7027	. . . . . . . . . . . . 13 ⊢ ((𝐺:𝐴⟶𝐶 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐶)
10		opelxpi 5661	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) ∈ 𝐵 ∧ (𝐺‘𝑥) ∈ 𝐶) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐶))
11	8, 9, 10	syl2an 597	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) ∧ (𝐺:𝐴⟶𝐶 ∧ 𝑥 ∈ 𝐴)) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐶))
12	11	anandirs 680	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ 𝑥 ∈ 𝐴) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐶))
13	12	ralrimiva 3130	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → ∀𝑥 ∈ 𝐴 ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐶))
14	13	3adant1 1131	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → ∀𝑥 ∈ 𝐴 ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐶))
15		eqid 2737	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
16	15	fmpt 7056	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐶) ↔ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩):𝐴⟶(𝐵 × 𝐶))
17	14, 16	sylib 218	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩):𝐴⟶(𝐵 × 𝐶))
18	17	ffnd 6663	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) Fn 𝐴)
19	18	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ (ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) → (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) Fn 𝐴)
20		xpss 5640	. . . . . . . . . . 11 ⊢ (𝐵 × 𝐶) ⊆ (V × V)
21		ffvelcdm 7027	. . . . . . . . . . 11 ⊢ ((ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝑧 ∈ 𝐴) → (ℎ‘𝑧) ∈ (𝐵 × 𝐶))
22	20, 21	sselid 3920	. . . . . . . . . 10 ⊢ ((ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝑧 ∈ 𝐴) → (ℎ‘𝑧) ∈ (V × V))
23	22	3ad2antl1 1187	. . . . . . . . 9 ⊢ (((ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)) ∧ 𝑧 ∈ 𝐴) → (ℎ‘𝑧) ∈ (V × V))
24	23	adantll 715	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ (ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) ∧ 𝑧 ∈ 𝐴) → (ℎ‘𝑧) ∈ (V × V))
25		fveq1 6833	. . . . . . . . . . . 12 ⊢ (𝐹 = (𝑃 ∘ ℎ) → (𝐹‘𝑧) = ((𝑃 ∘ ℎ)‘𝑧))
26		upxp.1	. . . . . . . . . . . . . 14 ⊢ 𝑃 = (1st ↾ (𝐵 × 𝐶))
27	26	coeq1i 5808	. . . . . . . . . . . . 13 ⊢ (𝑃 ∘ ℎ) = ((1st ↾ (𝐵 × 𝐶)) ∘ ℎ)
28	27	fveq1i 6835	. . . . . . . . . . . 12 ⊢ ((𝑃 ∘ ℎ)‘𝑧) = (((1st ↾ (𝐵 × 𝐶)) ∘ ℎ)‘𝑧)
29	25, 28	eqtrdi 2788	. . . . . . . . . . 11 ⊢ (𝐹 = (𝑃 ∘ ℎ) → (𝐹‘𝑧) = (((1st ↾ (𝐵 × 𝐶)) ∘ ℎ)‘𝑧))
30	29	3ad2ant2 1135	. . . . . . . . . 10 ⊢ ((ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)) → (𝐹‘𝑧) = (((1st ↾ (𝐵 × 𝐶)) ∘ ℎ)‘𝑧))
31	30	ad2antlr 728	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ (ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (((1st ↾ (𝐵 × 𝐶)) ∘ ℎ)‘𝑧))
32		simpr1 1196	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ (ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) → ℎ:𝐴⟶(𝐵 × 𝐶))
33		fvco3 6933	. . . . . . . . . 10 ⊢ ((ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝑧 ∈ 𝐴) → (((1st ↾ (𝐵 × 𝐶)) ∘ ℎ)‘𝑧) = ((1st ↾ (𝐵 × 𝐶))‘(ℎ‘𝑧)))
34	32, 33	sylan 581	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ (ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) ∧ 𝑧 ∈ 𝐴) → (((1st ↾ (𝐵 × 𝐶)) ∘ ℎ)‘𝑧) = ((1st ↾ (𝐵 × 𝐶))‘(ℎ‘𝑧)))
35	21	3ad2antl1 1187	. . . . . . . . . . 11 ⊢ (((ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)) ∧ 𝑧 ∈ 𝐴) → (ℎ‘𝑧) ∈ (𝐵 × 𝐶))
36	35	adantll 715	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ (ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) ∧ 𝑧 ∈ 𝐴) → (ℎ‘𝑧) ∈ (𝐵 × 𝐶))
37	36	fvresd 6854	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ (ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) ∧ 𝑧 ∈ 𝐴) → ((1st ↾ (𝐵 × 𝐶))‘(ℎ‘𝑧)) = (1st ‘(ℎ‘𝑧)))
38	31, 34, 37	3eqtrrd 2777	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ (ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) ∧ 𝑧 ∈ 𝐴) → (1st ‘(ℎ‘𝑧)) = (𝐹‘𝑧))
39		fveq1 6833	. . . . . . . . . . . 12 ⊢ (𝐺 = (𝑄 ∘ ℎ) → (𝐺‘𝑧) = ((𝑄 ∘ ℎ)‘𝑧))
40		upxp.2	. . . . . . . . . . . . . 14 ⊢ 𝑄 = (2nd ↾ (𝐵 × 𝐶))
41	40	coeq1i 5808	. . . . . . . . . . . . 13 ⊢ (𝑄 ∘ ℎ) = ((2nd ↾ (𝐵 × 𝐶)) ∘ ℎ)
42	41	fveq1i 6835	. . . . . . . . . . . 12 ⊢ ((𝑄 ∘ ℎ)‘𝑧) = (((2nd ↾ (𝐵 × 𝐶)) ∘ ℎ)‘𝑧)
43	39, 42	eqtrdi 2788	. . . . . . . . . . 11 ⊢ (𝐺 = (𝑄 ∘ ℎ) → (𝐺‘𝑧) = (((2nd ↾ (𝐵 × 𝐶)) ∘ ℎ)‘𝑧))
44	43	3ad2ant3 1136	. . . . . . . . . 10 ⊢ ((ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)) → (𝐺‘𝑧) = (((2nd ↾ (𝐵 × 𝐶)) ∘ ℎ)‘𝑧))
45	44	ad2antlr 728	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ (ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) ∧ 𝑧 ∈ 𝐴) → (𝐺‘𝑧) = (((2nd ↾ (𝐵 × 𝐶)) ∘ ℎ)‘𝑧))
46		fvco3 6933	. . . . . . . . . 10 ⊢ ((ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝑧 ∈ 𝐴) → (((2nd ↾ (𝐵 × 𝐶)) ∘ ℎ)‘𝑧) = ((2nd ↾ (𝐵 × 𝐶))‘(ℎ‘𝑧)))
47	32, 46	sylan 581	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ (ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) ∧ 𝑧 ∈ 𝐴) → (((2nd ↾ (𝐵 × 𝐶)) ∘ ℎ)‘𝑧) = ((2nd ↾ (𝐵 × 𝐶))‘(ℎ‘𝑧)))
48	36	fvresd 6854	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ (ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) ∧ 𝑧 ∈ 𝐴) → ((2nd ↾ (𝐵 × 𝐶))‘(ℎ‘𝑧)) = (2nd ‘(ℎ‘𝑧)))
49	45, 47, 48	3eqtrrd 2777	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ (ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) ∧ 𝑧 ∈ 𝐴) → (2nd ‘(ℎ‘𝑧)) = (𝐺‘𝑧))
50		eqopi 7971	. . . . . . . 8 ⊢ (((ℎ‘𝑧) ∈ (V × V) ∧ ((1st ‘(ℎ‘𝑧)) = (𝐹‘𝑧) ∧ (2nd ‘(ℎ‘𝑧)) = (𝐺‘𝑧))) → (ℎ‘𝑧) = ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩)
51	24, 38, 49, 50	syl12anc 837	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ (ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) ∧ 𝑧 ∈ 𝐴) → (ℎ‘𝑧) = ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩)
52		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
53		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑥) = (𝐺‘𝑧))
54	52, 53	opeq12d 4825	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ = ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩)
55		opex 5411	. . . . . . . . 9 ⊢ ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩ ∈ V
56	54, 15, 55	fvmpt 6941	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑧) = ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩)
57	56	adantl 481	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ (ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) ∧ 𝑧 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑧) = ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩)
58	51, 57	eqtr4d 2775	. . . . . 6 ⊢ ((((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ (ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) ∧ 𝑧 ∈ 𝐴) → (ℎ‘𝑧) = ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑧))
59	7, 19, 58	eqfnfvd 6980	. . . . 5 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ (ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) → ℎ = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))
60	59	ex 412	. . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → ((ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)) → ℎ = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
61		ffn 6662	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
62	61	3ad2ant2 1135	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → 𝐹 Fn 𝐴)
63		fo1st 7955	. . . . . . . . . . 11 ⊢ 1st :V–onto→V
64		fofn 6748	. . . . . . . . . . 11 ⊢ (1st :V–onto→V → 1st Fn V)
65	63, 64	ax-mp 5	. . . . . . . . . 10 ⊢ 1st Fn V
66		ssv 3947	. . . . . . . . . 10 ⊢ (𝐵 × 𝐶) ⊆ V
67		fnssres 6615	. . . . . . . . . 10 ⊢ ((1st Fn V ∧ (𝐵 × 𝐶) ⊆ V) → (1st ↾ (𝐵 × 𝐶)) Fn (𝐵 × 𝐶))
68	65, 66, 67	mp2an 693	. . . . . . . . 9 ⊢ (1st ↾ (𝐵 × 𝐶)) Fn (𝐵 × 𝐶)
69	17	frnd 6670	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → ran (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ⊆ (𝐵 × 𝐶))
70		fnco 6610	. . . . . . . . 9 ⊢ (((1st ↾ (𝐵 × 𝐶)) Fn (𝐵 × 𝐶) ∧ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) Fn 𝐴 ∧ ran (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ⊆ (𝐵 × 𝐶)) → ((1st ↾ (𝐵 × 𝐶)) ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) Fn 𝐴)
71	68, 18, 69, 70	mp3an2i 1469	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → ((1st ↾ (𝐵 × 𝐶)) ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) Fn 𝐴)
72		fvco3 6933	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩):𝐴⟶(𝐵 × 𝐶) ∧ 𝑧 ∈ 𝐴) → (((1st ↾ (𝐵 × 𝐶)) ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))‘𝑧) = ((1st ↾ (𝐵 × 𝐶))‘((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑧)))
73	17, 72	sylan 581	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ 𝑧 ∈ 𝐴) → (((1st ↾ (𝐵 × 𝐶)) ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))‘𝑧) = ((1st ↾ (𝐵 × 𝐶))‘((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑧)))
74	56	adantl 481	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ 𝑧 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑧) = ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩)
75	74	fveq2d 6838	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ 𝑧 ∈ 𝐴) → ((1st ↾ (𝐵 × 𝐶))‘((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑧)) = ((1st ↾ (𝐵 × 𝐶))‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩))
76		ffvelcdm 7027	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝐵)
77		ffvelcdm 7027	. . . . . . . . . . . . . 14 ⊢ ((𝐺:𝐴⟶𝐶 ∧ 𝑧 ∈ 𝐴) → (𝐺‘𝑧) ∈ 𝐶)
78		opelxpi 5661	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑧) ∈ 𝐵 ∧ (𝐺‘𝑧) ∈ 𝐶) → ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩ ∈ (𝐵 × 𝐶))
79	76, 77, 78	syl2an 597	. . . . . . . . . . . . 13 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) ∧ (𝐺:𝐴⟶𝐶 ∧ 𝑧 ∈ 𝐴)) → ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩ ∈ (𝐵 × 𝐶))
80	79	anandirs 680	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ 𝑧 ∈ 𝐴) → ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩ ∈ (𝐵 × 𝐶))
81	80	3adantl1 1168	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ 𝑧 ∈ 𝐴) → ⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩ ∈ (𝐵 × 𝐶))
82	81	fvresd 6854	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ 𝑧 ∈ 𝐴) → ((1st ↾ (𝐵 × 𝐶))‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩) = (1st ‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩))
83		fvex 6847	. . . . . . . . . . 11 ⊢ (𝐹‘𝑧) ∈ V
84		fvex 6847	. . . . . . . . . . 11 ⊢ (𝐺‘𝑧) ∈ V
85	83, 84	op1st 7943	. . . . . . . . . 10 ⊢ (1st ‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩) = (𝐹‘𝑧)
86	82, 85	eqtrdi 2788	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ 𝑧 ∈ 𝐴) → ((1st ↾ (𝐵 × 𝐶))‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩) = (𝐹‘𝑧))
87	73, 75, 86	3eqtrrd 2777	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (((1st ↾ (𝐵 × 𝐶)) ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))‘𝑧))
88	62, 71, 87	eqfnfvd 6980	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → 𝐹 = ((1st ↾ (𝐵 × 𝐶)) ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
89	26	coeq1i 5808	. . . . . . 7 ⊢ (𝑃 ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) = ((1st ↾ (𝐵 × 𝐶)) ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))
90	88, 89	eqtr4di 2790	. . . . . 6 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → 𝐹 = (𝑃 ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
91		ffn 6662	. . . . . . . . 9 ⊢ (𝐺:𝐴⟶𝐶 → 𝐺 Fn 𝐴)
92	91	3ad2ant3 1136	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → 𝐺 Fn 𝐴)
93		fo2nd 7956	. . . . . . . . . . 11 ⊢ 2nd :V–onto→V
94		fofn 6748	. . . . . . . . . . 11 ⊢ (2nd :V–onto→V → 2nd Fn V)
95	93, 94	ax-mp 5	. . . . . . . . . 10 ⊢ 2nd Fn V
96		fnssres 6615	. . . . . . . . . 10 ⊢ ((2nd Fn V ∧ (𝐵 × 𝐶) ⊆ V) → (2nd ↾ (𝐵 × 𝐶)) Fn (𝐵 × 𝐶))
97	95, 66, 96	mp2an 693	. . . . . . . . 9 ⊢ (2nd ↾ (𝐵 × 𝐶)) Fn (𝐵 × 𝐶)
98		fnco 6610	. . . . . . . . 9 ⊢ (((2nd ↾ (𝐵 × 𝐶)) Fn (𝐵 × 𝐶) ∧ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) Fn 𝐴 ∧ ran (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ⊆ (𝐵 × 𝐶)) → ((2nd ↾ (𝐵 × 𝐶)) ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) Fn 𝐴)
99	97, 18, 69, 98	mp3an2i 1469	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → ((2nd ↾ (𝐵 × 𝐶)) ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) Fn 𝐴)
100		fvco3 6933	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩):𝐴⟶(𝐵 × 𝐶) ∧ 𝑧 ∈ 𝐴) → (((2nd ↾ (𝐵 × 𝐶)) ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))‘𝑧) = ((2nd ↾ (𝐵 × 𝐶))‘((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑧)))
101	17, 100	sylan 581	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ 𝑧 ∈ 𝐴) → (((2nd ↾ (𝐵 × 𝐶)) ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))‘𝑧) = ((2nd ↾ (𝐵 × 𝐶))‘((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑧)))
102	74	fveq2d 6838	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ 𝑧 ∈ 𝐴) → ((2nd ↾ (𝐵 × 𝐶))‘((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑧)) = ((2nd ↾ (𝐵 × 𝐶))‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩))
103	81	fvresd 6854	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ 𝑧 ∈ 𝐴) → ((2nd ↾ (𝐵 × 𝐶))‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩) = (2nd ‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩))
104	83, 84	op2nd 7944	. . . . . . . . . 10 ⊢ (2nd ‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩) = (𝐺‘𝑧)
105	103, 104	eqtrdi 2788	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ 𝑧 ∈ 𝐴) → ((2nd ↾ (𝐵 × 𝐶))‘⟨(𝐹‘𝑧), (𝐺‘𝑧)⟩) = (𝐺‘𝑧))
106	101, 102, 105	3eqtrrd 2777	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) ∧ 𝑧 ∈ 𝐴) → (𝐺‘𝑧) = (((2nd ↾ (𝐵 × 𝐶)) ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))‘𝑧))
107	92, 99, 106	eqfnfvd 6980	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → 𝐺 = ((2nd ↾ (𝐵 × 𝐶)) ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
108	40	coeq1i 5808	. . . . . . 7 ⊢ (𝑄 ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) = ((2nd ↾ (𝐵 × 𝐶)) ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))
109	107, 108	eqtr4di 2790	. . . . . 6 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → 𝐺 = (𝑄 ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
110	17, 90, 109	3jca 1129	. . . . 5 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩):𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) ∧ 𝐺 = (𝑄 ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))))
111		feq1 6640	. . . . . 6 ⊢ (ℎ = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) → (ℎ:𝐴⟶(𝐵 × 𝐶) ↔ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩):𝐴⟶(𝐵 × 𝐶)))
112		coeq2 5807	. . . . . . 7 ⊢ (ℎ = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) → (𝑃 ∘ ℎ) = (𝑃 ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
113	112	eqeq2d 2748	. . . . . 6 ⊢ (ℎ = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) → (𝐹 = (𝑃 ∘ ℎ) ↔ 𝐹 = (𝑃 ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))))
114		coeq2 5807	. . . . . . 7 ⊢ (ℎ = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) → (𝑄 ∘ ℎ) = (𝑄 ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
115	114	eqeq2d 2748	. . . . . 6 ⊢ (ℎ = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) → (𝐺 = (𝑄 ∘ ℎ) ↔ 𝐺 = (𝑄 ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))))
116	111, 113, 115	3anbi123d 1439	. . . . 5 ⊢ (ℎ = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) → ((ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)) ↔ ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩):𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)) ∧ 𝐺 = (𝑄 ∘ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))))
117	110, 116	syl5ibrcom 247	. . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → (ℎ = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) → (ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))))
118	60, 117	impbid 212	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → ((ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)) ↔ ℎ = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
119	118	eubidv 2587	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → (∃!ℎ(ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)) ↔ ∃!ℎ ℎ = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)))
120	4, 119	mpbird 257	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → ∃!ℎ(ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃!weu 2569  ∀wral 3052  Vcvv 3430   ⊆ wss 3890  ⟨cop 4574   ↦ cmpt 5167   × cxp 5622  ran crn 5625   ↾ cres 5626   ∘ ccom 5628   Fn wfn 6487  ⟶wf 6488  –onto→wfo 6490  ‘cfv 6492  1^st c1st 7933  2^nd c2nd 7934

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-1st 7935  df-2nd 7936

This theorem is referenced by:  uptx  23600  txcn  23601