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Theorem ucnimalem 24310

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 3492	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3492	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	op1std 8040	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝑝) = 𝑥)
5	4	fveq2d 6924	. . . 4 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1^st ‘𝑝)) = (𝐹‘𝑥))
6	2, 3	op2ndd 8041	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝑝) = 𝑦)
7	6	fveq2d 6924	. . . 4 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2^nd ‘𝑝)) = (𝐹‘𝑦))
8	5, 7	opeq12d 4905	. . 3 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ⟨(𝐹‘(1^st ‘𝑝)), (𝐹‘(2^nd ‘𝑝))⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
9	8	mpompt 7564	. 2 ⊢ (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1^st ‘𝑝)), (𝐹‘(2^nd ‘𝑝))⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
10	1, 9	eqtr4i 2771	1 ⊢ 𝐺 = (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1^st ‘𝑝)), (𝐹‘(2^nd ‘𝑝))⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 ⟨cop 4654 ↦ cmpt 5249 × cxp 5698 ‘cfv 6573 (class class class)co 7448 ∈ cmpo 7450 1^st c1st 8028 2^nd c2nd 8029 UnifOncust 24229 Cn_ucucn 24305

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-iota 6525 df-fun 6575 df-fv 6581 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031

This theorem is referenced by: ucnima 24311

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