dvhvscacbv - Mathbox for Norm Megill

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Theorem dvhvscacbv 39634

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 6846	. . . 4 ⊢ (𝑠 = 𝑡 → (𝑠‘(1^st ‘𝑓)) = (𝑡‘(1^st ‘𝑓)))
3		coeq1 5818	. . . 4 ⊢ (𝑠 = 𝑡 → (𝑠 ∘ (2^nd ‘𝑓)) = (𝑡 ∘ (2^nd ‘𝑓)))
4	2, 3	opeq12d 4843	. . 3 ⊢ (𝑠 = 𝑡 → ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩ = ⟨(𝑡‘(1^st ‘𝑓)), (𝑡 ∘ (2^nd ‘𝑓))⟩)
5		2fveq3 6852	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑡‘(1^st ‘𝑓)) = (𝑡‘(1^st ‘𝑔)))
6		fveq2 6847	. . . . 5 ⊢ (𝑓 = 𝑔 → (2^nd ‘𝑓) = (2^nd ‘𝑔))
7	6	coeq2d 5823	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑡 ∘ (2^nd ‘𝑓)) = (𝑡 ∘ (2^nd ‘𝑔)))
8	5, 7	opeq12d 4843	. . 3 ⊢ (𝑓 = 𝑔 → ⟨(𝑡‘(1^st ‘𝑓)), (𝑡 ∘ (2^nd ‘𝑓))⟩ = ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩)
9	4, 8	cbvmpov 7457	. 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 2759	1 ⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩)

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ⟨cop 4597 × cxp 5636 ∘ ccom 5642 ‘cfv 6501 ∈ cmpo 7364 1^st c1st 7924 2^nd c2nd 7925

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5261 ax-nul 5268 ax-pr 5389

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-rab 3406 df-v 3448 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-co 5647 df-iota 6453 df-fv 6509 df-oprab 7366 df-mpo 7367

This theorem is referenced by: dvhvscaval 39635