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Theorem ucnimalem 24305

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 3482	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3482	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	op1std 8023	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝑝) = 𝑥)
5	4	fveq2d 6911	. . . 4 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1^st ‘𝑝)) = (𝐹‘𝑥))
6	2, 3	op2ndd 8024	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝑝) = 𝑦)
7	6	fveq2d 6911	. . . 4 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2^nd ‘𝑝)) = (𝐹‘𝑦))
8	5, 7	opeq12d 4886	. . 3 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ⟨(𝐹‘(1^st ‘𝑝)), (𝐹‘(2^nd ‘𝑝))⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
9	8	mpompt 7547	. 2 ⊢ (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1^st ‘𝑝)), (𝐹‘(2^nd ‘𝑝))⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
10	1, 9	eqtr4i 2766	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 2106 ⟨cop 4637 ↦ cmpt 5231 × cxp 5687 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 1^st c1st 8011 2^nd c2nd 8012 UnifOncust 24224 Cn_ucucn 24300

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014

This theorem is referenced by: ucnima 24306

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