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Theorem dvhvscacbv 36704
Description: Change bound variables to isolate them later. (Contributed by NM, 20-Nov-2013.)
Hypothesis
Ref Expression
dvhvscaval.s · = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
Assertion
Ref Expression
dvhvscacbv · = (𝑡𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
Distinct variable groups:   𝑓,𝑠,𝑡,𝑔,𝐸   𝑇,𝑠,𝑓,𝑡,𝑔
Allowed substitution hints:   · (𝑡,𝑓,𝑔,𝑠)

Proof of Theorem dvhvscacbv
StepHypRef Expression
1 dvhvscaval.s . 2 · = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
2 fveq1 6228 . . . 4 (𝑠 = 𝑡 → (𝑠‘(1st𝑓)) = (𝑡‘(1st𝑓)))
3 coeq1 5312 . . . 4 (𝑠 = 𝑡 → (𝑠 ∘ (2nd𝑓)) = (𝑡 ∘ (2nd𝑓)))
42, 3opeq12d 4441 . . 3 (𝑠 = 𝑡 → ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩ = ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩)
5 fveq2 6229 . . . . 5 (𝑓 = 𝑔 → (1st𝑓) = (1st𝑔))
65fveq2d 6233 . . . 4 (𝑓 = 𝑔 → (𝑡‘(1st𝑓)) = (𝑡‘(1st𝑔)))
7 fveq2 6229 . . . . 5 (𝑓 = 𝑔 → (2nd𝑓) = (2nd𝑔))
87coeq2d 5317 . . . 4 (𝑓 = 𝑔 → (𝑡 ∘ (2nd𝑓)) = (𝑡 ∘ (2nd𝑔)))
96, 8opeq12d 4441 . . 3 (𝑓 = 𝑔 → ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩ = ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
104, 9cbvmpt2v 6777 . 2 (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩) = (𝑡𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
111, 10eqtri 2673 1 · = (𝑡𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  cop 4216   × cxp 5141  ccom 5147  cfv 5926  cmpt2 6692  1st c1st 7208  2nd c2nd 7209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-co 5152  df-iota 5889  df-fv 5934  df-oprab 6694  df-mpt2 6695
This theorem is referenced by:  dvhvscaval  36705
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