Theorem dvhvscacbv 41727

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 6866	. . . 4 ⊢ (𝑠 = 𝑡 → (𝑠‘(1st ‘𝑓)) = (𝑡‘(1st ‘𝑓)))
3		coeq1 5830	. . . 4 ⊢ (𝑠 = 𝑡 → (𝑠 ∘ (2nd ‘𝑓)) = (𝑡 ∘ (2nd ‘𝑓)))
4	2, 3	opeq12d 4840	. . 3 ⊢ (𝑠 = 𝑡 → ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩ = ⟨(𝑡‘(1st ‘𝑓)), (𝑡 ∘ (2nd ‘𝑓))⟩)
5		2fveq3 6872	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑡‘(1st ‘𝑓)) = (𝑡‘(1st ‘𝑔)))
6		fveq2 6867	. . . . 5 ⊢ (𝑓 = 𝑔 → (2nd ‘𝑓) = (2nd ‘𝑔))
7	6	coeq2d 5835	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑡 ∘ (2nd ‘𝑓)) = (𝑡 ∘ (2nd ‘𝑔)))
8	5, 7	opeq12d 4840	. . 3 ⊢ (𝑓 = 𝑔 → ⟨(𝑡‘(1st ‘𝑓)), (𝑡 ∘ (2nd ‘𝑓))⟩ = ⟨(𝑡‘(1st ‘𝑔)), (𝑡 ∘ (2nd ‘𝑔))⟩)
9	4, 8	cbvmpov 7491	. 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 2786	1 ⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st ‘𝑔)), (𝑡 ∘ (2nd ‘𝑔))⟩)

Syntax hints:   = wceq 1561  ⟨cop 4589   × cxp 5646   ∘ ccom 5652  ‘cfv 6521   ∈ cmpo 7398  1^st c1st 7968  2^nd c2nd 7969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-co 5657  df-iota 6477  df-fv 6529  df-oprab 7400  df-mpo 7401

This theorem is referenced by:  dvhvscaval  41728