dvhvscacbv - Mathbox for Norm Megill

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Theorem dvhvscacbv 35405

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 6102	. . . 4 ⊢ (𝑠 = 𝑡 → (𝑠‘(1^st ‘𝑓)) = (𝑡‘(1^st ‘𝑓)))
3		coeq1 5201	. . . 4 ⊢ (𝑠 = 𝑡 → (𝑠 ∘ (2^nd ‘𝑓)) = (𝑡 ∘ (2^nd ‘𝑓)))
4	2, 3	opeq12d 4348	. . 3 ⊢ (𝑠 = 𝑡 → ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩ = ⟨(𝑡‘(1^st ‘𝑓)), (𝑡 ∘ (2^nd ‘𝑓))⟩)
5		fveq2 6103	. . . . 5 ⊢ (𝑓 = 𝑔 → (1^st ‘𝑓) = (1^st ‘𝑔))
6	5	fveq2d 6107	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑡‘(1^st ‘𝑓)) = (𝑡‘(1^st ‘𝑔)))
7		fveq2 6103	. . . . 5 ⊢ (𝑓 = 𝑔 → (2^nd ‘𝑓) = (2^nd ‘𝑔))
8	7	coeq2d 5206	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑡 ∘ (2^nd ‘𝑓)) = (𝑡 ∘ (2^nd ‘𝑔)))
9	6, 8	opeq12d 4348	. . 3 ⊢ (𝑓 = 𝑔 → ⟨(𝑡‘(1^st ‘𝑓)), (𝑡 ∘ (2^nd ‘𝑓))⟩ = ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩)
10	4, 9	cbvmpt2v 6633	. 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))⟩)
11	1, 10	eqtri 2632	1 ⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩)

Colors of variables: wff setvar class

Syntax hints: = wceq 1475 ⟨cop 4131 × cxp 5036 ∘ ccom 5042 ‘cfv 5804 ↦ cmpt2 6551 1^st c1st 7057 2^nd c2nd 7058

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-co 5047 df-iota 5768 df-fv 5812 df-oprab 6553 df-mpt2 6554

This theorem is referenced by: dvhvscaval 35406