dvhvscacbv - Mathbox for Norm Megill

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Theorem dvhvscacbv 38226

Description: Change bound variables to isolate them later. (Contributed by NM, 20-Nov-2013.)

**Hypothesis**
Ref	Expression
dvhvscaval.s	⊢ · = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)

**Assertion**
Ref	Expression
dvhvscacbv	⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩)

Distinct variable groups: 𝑓,𝑠,𝑡,𝑔,𝐸 𝑇,𝑠,𝑓,𝑡,𝑔 Allowed substitution hints: · (𝑡,𝑓,𝑔,𝑠)

**Proof of Theorem dvhvscacbv**
Step	Hyp	Ref	Expression
1		dvhvscaval.s	. 2 ⊢ · = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)
2		fveq1 6662	. . . 4 ⊢ (𝑠 = 𝑡 → (𝑠‘(1^st ‘𝑓)) = (𝑡‘(1^st ‘𝑓)))
3		coeq1 5721	. . . 4 ⊢ (𝑠 = 𝑡 → (𝑠 ∘ (2^nd ‘𝑓)) = (𝑡 ∘ (2^nd ‘𝑓)))
4	2, 3	opeq12d 4803	. . 3 ⊢ (𝑠 = 𝑡 → ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩ = ⟨(𝑡‘(1^st ‘𝑓)), (𝑡 ∘ (2^nd ‘𝑓))⟩)
5		2fveq3 6668	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑡‘(1^st ‘𝑓)) = (𝑡‘(1^st ‘𝑔)))
6		fveq2 6663	. . . . 5 ⊢ (𝑓 = 𝑔 → (2^nd ‘𝑓) = (2^nd ‘𝑔))
7	6	coeq2d 5726	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑡 ∘ (2^nd ‘𝑓)) = (𝑡 ∘ (2^nd ‘𝑔)))
8	5, 7	opeq12d 4803	. . 3 ⊢ (𝑓 = 𝑔 → ⟨(𝑡‘(1^st ‘𝑓)), (𝑡 ∘ (2^nd ‘𝑓))⟩ = ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩)
9	4, 8	cbvmpov 7241	. 2 ⊢ (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩) = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩)
10	1, 9	eqtri 2842	1 ⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩)

Colors of variables: wff setvar class

Syntax hints: = wceq 1531 ⟨cop 4565 × cxp 5546 ∘ ccom 5552 ‘cfv 6348 ∈ cmpo 7150 1^st c1st 7679 2^nd c2nd 7680

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-rex 3142 df-rab 3145 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-co 5557 df-iota 6307 df-fv 6356 df-oprab 7152 df-mpo 7153

This theorem is referenced by: dvhvscaval 38227