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Theorem ucnimalem 24290

Description: Reformulate the 𝐺 function as a mapping with one variable. (Contributed by Thierry Arnoux, 19-Nov-2017.)

**Hypotheses**
Ref	Expression
ucnprima.1	⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋))
ucnprima.2	⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌))
ucnprima.3	⊢ (𝜑 → 𝐹 ∈ (𝑈 Cn_u𝑉))
ucnprima.4	⊢ (𝜑 → 𝑊 ∈ 𝑉)
ucnprima.5	⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)

**Assertion**
Ref	Expression
ucnimalem	⊢ 𝐺 = (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1^st ‘𝑝)), (𝐹‘(2^nd ‘𝑝))⟩)

Distinct variable groups: 𝑥,𝑝,𝑦,𝐹 𝑋,𝑝,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑝) 𝑈(𝑥,𝑦,𝑝) 𝐺(𝑥,𝑦,𝑝) 𝑉(𝑥,𝑦,𝑝) 𝑊(𝑥,𝑦,𝑝) 𝑌(𝑥,𝑦,𝑝)

**Proof of Theorem ucnimalem**
Step	Hyp	Ref	Expression
1		ucnprima.5	. 2 ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
2		vex 3483	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3483	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	op1std 8025	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝑝) = 𝑥)
5	4	fveq2d 6909	. . . 4 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1^st ‘𝑝)) = (𝐹‘𝑥))
6	2, 3	op2ndd 8026	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝑝) = 𝑦)
7	6	fveq2d 6909	. . . 4 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2^nd ‘𝑝)) = (𝐹‘𝑦))
8	5, 7	opeq12d 4880	. . 3 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ⟨(𝐹‘(1^st ‘𝑝)), (𝐹‘(2^nd ‘𝑝))⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
9	8	mpompt 7548	. 2 ⊢ (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1^st ‘𝑝)), (𝐹‘(2^nd ‘𝑝))⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
10	1, 9	eqtr4i 2767	1 ⊢ 𝐺 = (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1^st ‘𝑝)), (𝐹‘(2^nd ‘𝑝))⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ⟨cop 4631 ↦ cmpt 5224 × cxp 5682 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 1^st c1st 8013 2^nd c2nd 8014 UnifOncust 24209 Cn_ucucn 24285

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fv 6568 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016

This theorem is referenced by: ucnima 24291

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