Theorem dvhvscacbv 39969

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‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)

Assertion

Ref	Expression
dvhvscacbv	⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st ‘𝑔)), (𝑡 ∘ (2nd ‘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‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)
2		fveq1 6891	. . . 4 ⊢ (𝑠 = 𝑡 → (𝑠‘(1st ‘𝑓)) = (𝑡‘(1st ‘𝑓)))
3		coeq1 5858	. . . 4 ⊢ (𝑠 = 𝑡 → (𝑠 ∘ (2nd ‘𝑓)) = (𝑡 ∘ (2nd ‘𝑓)))
4	2, 3	opeq12d 4882	. . 3 ⊢ (𝑠 = 𝑡 → ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩ = ⟨(𝑡‘(1st ‘𝑓)), (𝑡 ∘ (2nd ‘𝑓))⟩)
5		2fveq3 6897	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑡‘(1st ‘𝑓)) = (𝑡‘(1st ‘𝑔)))
6		fveq2 6892	. . . . 5 ⊢ (𝑓 = 𝑔 → (2nd ‘𝑓) = (2nd ‘𝑔))
7	6	coeq2d 5863	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑡 ∘ (2nd ‘𝑓)) = (𝑡 ∘ (2nd ‘𝑔)))
8	5, 7	opeq12d 4882	. . 3 ⊢ (𝑓 = 𝑔 → ⟨(𝑡‘(1st ‘𝑓)), (𝑡 ∘ (2nd ‘𝑓))⟩ = ⟨(𝑡‘(1st ‘𝑔)), (𝑡 ∘ (2nd ‘𝑔))⟩)
9	4, 8	cbvmpov 7504	. 2 ⊢ (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩) = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st ‘𝑔)), (𝑡 ∘ (2nd ‘𝑔))⟩)
10	1, 9	eqtri 2761	1 ⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st ‘𝑔)), (𝑡 ∘ (2nd ‘𝑔))⟩)

Syntax hints:   = wceq 1542  ⟨cop 4635   × cxp 5675   ∘ ccom 5681  ‘cfv 6544   ∈ cmpo 7411  1^st c1st 7973  2^nd c2nd 7974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-co 5686  df-iota 6496  df-fv 6552  df-oprab 7413  df-mpo 7414

This theorem is referenced by:  dvhvscaval  39970