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Theorem ucnimalem 24222

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 3434	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3434	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	op1std 7943	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝑝) = 𝑥)
5	4	fveq2d 6836	. . . 4 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1^st ‘𝑝)) = (𝐹‘𝑥))
6	2, 3	op2ndd 7944	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝑝) = 𝑦)
7	6	fveq2d 6836	. . . 4 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2^nd ‘𝑝)) = (𝐹‘𝑦))
8	5, 7	opeq12d 4825	. . 3 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ⟨(𝐹‘(1^st ‘𝑝)), (𝐹‘(2^nd ‘𝑝))⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
9	8	mpompt 7472	. 2 ⊢ (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1^st ‘𝑝)), (𝐹‘(2^nd ‘𝑝))⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
10	1, 9	eqtr4i 2763	1 ⊢ 𝐺 = (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1^st ‘𝑝)), (𝐹‘(2^nd ‘𝑝))⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⟨cop 4574 ↦ cmpt 5167 × cxp 5620 ‘cfv 6490 (class class class)co 7358 ∈ cmpo 7360 1^st c1st 7931 2^nd c2nd 7932 UnifOncust 24143 Cn_ucucn 24217

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-sep 5231 ax-nul 5241 ax-pr 5368 ax-un 7680

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-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 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-iota 6446 df-fun 6492 df-fv 6498 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934

This theorem is referenced by: ucnima 24223

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