Theorem dvhvscacbv 41057

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 6921	. . . 4 ⊢ (𝑠 = 𝑡 → (𝑠‘(1st ‘𝑓)) = (𝑡‘(1st ‘𝑓)))
3		coeq1 5882	. . . 4 ⊢ (𝑠 = 𝑡 → (𝑠 ∘ (2nd ‘𝑓)) = (𝑡 ∘ (2nd ‘𝑓)))
4	2, 3	opeq12d 4905	. . 3 ⊢ (𝑠 = 𝑡 → ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩ = ⟨(𝑡‘(1st ‘𝑓)), (𝑡 ∘ (2nd ‘𝑓))⟩)
5		2fveq3 6927	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑡‘(1st ‘𝑓)) = (𝑡‘(1st ‘𝑔)))
6		fveq2 6922	. . . . 5 ⊢ (𝑓 = 𝑔 → (2nd ‘𝑓) = (2nd ‘𝑔))
7	6	coeq2d 5887	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑡 ∘ (2nd ‘𝑓)) = (𝑡 ∘ (2nd ‘𝑔)))
8	5, 7	opeq12d 4905	. . 3 ⊢ (𝑓 = 𝑔 → ⟨(𝑡‘(1st ‘𝑓)), (𝑡 ∘ (2nd ‘𝑓))⟩ = ⟨(𝑡‘(1st ‘𝑔)), (𝑡 ∘ (2nd ‘𝑔))⟩)
9	4, 8	cbvmpov 7547	. 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 2768	1 ⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st ‘𝑔)), (𝑡 ∘ (2nd ‘𝑔))⟩)

Syntax hints:   = wceq 1537  ⟨cop 4654   × cxp 5698   ∘ ccom 5704  ‘cfv 6575   ∈ cmpo 7452  1^st c1st 8030  2^nd c2nd 8031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-co 5709  df-iota 6527  df-fv 6583  df-oprab 7454  df-mpo 7455

This theorem is referenced by:  dvhvscaval  41058