dvhvscacbv - Mathbox for Norm Megill

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Theorem dvhvscacbv 37118

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 6411	. . . 4 ⊢ (𝑠 = 𝑡 → (𝑠‘(1^st ‘𝑓)) = (𝑡‘(1^st ‘𝑓)))
3		coeq1 5484	. . . 4 ⊢ (𝑠 = 𝑡 → (𝑠 ∘ (2^nd ‘𝑓)) = (𝑡 ∘ (2^nd ‘𝑓)))
4	2, 3	opeq12d 4602	. . 3 ⊢ (𝑠 = 𝑡 → ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩ = ⟨(𝑡‘(1^st ‘𝑓)), (𝑡 ∘ (2^nd ‘𝑓))⟩)
5		2fveq3 6417	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑡‘(1^st ‘𝑓)) = (𝑡‘(1^st ‘𝑔)))
6		fveq2 6412	. . . . 5 ⊢ (𝑓 = 𝑔 → (2^nd ‘𝑓) = (2^nd ‘𝑔))
7	6	coeq2d 5489	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑡 ∘ (2^nd ‘𝑓)) = (𝑡 ∘ (2^nd ‘𝑔)))
8	5, 7	opeq12d 4602	. . 3 ⊢ (𝑓 = 𝑔 → ⟨(𝑡‘(1^st ‘𝑓)), (𝑡 ∘ (2^nd ‘𝑓))⟩ = ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩)
9	4, 8	cbvmpt2v 6970	. 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 2822	1 ⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩)

Colors of variables: wff setvar class

Syntax hints: = wceq 1653 ⟨cop 4375 × cxp 5311 ∘ ccom 5317 ‘cfv 6102 ↦ cmpt2 6881 1^st c1st 7400 2^nd c2nd 7401

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pr 5098

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-rex 3096 df-rab 3099 df-v 3388 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-br 4845 df-opab 4907 df-co 5322 df-iota 6065 df-fv 6110 df-oprab 6883 df-mpt2 6884

This theorem is referenced by: dvhvscaval 37119