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Theorem ucnimalem 22886

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 3444	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3444	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	op1std 7681	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝑝) = 𝑥)
5	4	fveq2d 6649	. . . 4 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1^st ‘𝑝)) = (𝐹‘𝑥))
6	2, 3	op2ndd 7682	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝑝) = 𝑦)
7	6	fveq2d 6649	. . . 4 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2^nd ‘𝑝)) = (𝐹‘𝑦))
8	5, 7	opeq12d 4773	. . 3 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ⟨(𝐹‘(1^st ‘𝑝)), (𝐹‘(2^nd ‘𝑝))⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
9	8	mpompt 7245	. 2 ⊢ (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1^st ‘𝑝)), (𝐹‘(2^nd ‘𝑝))⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
10	1, 9	eqtr4i 2824	1 ⊢ 𝐺 = (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1^st ‘𝑝)), (𝐹‘(2^nd ‘𝑝))⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ⟨cop 4531 ↦ cmpt 5110 × cxp 5517 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 1^st c1st 7669 2^nd c2nd 7670 UnifOncust 22805 Cn_ucucn 22881

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672

This theorem is referenced by: ucnima 22887

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