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Theorem dvhvscacbv 41395
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 6834 . . . 4 (𝑠 = 𝑡 → (𝑠‘(1st𝑓)) = (𝑡‘(1st𝑓)))
3 coeq1 5807 . . . 4 (𝑠 = 𝑡 → (𝑠 ∘ (2nd𝑓)) = (𝑡 ∘ (2nd𝑓)))
42, 3opeq12d 4838 . . 3 (𝑠 = 𝑡 → ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩ = ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩)
5 2fveq3 6840 . . . 4 (𝑓 = 𝑔 → (𝑡‘(1st𝑓)) = (𝑡‘(1st𝑔)))
6 fveq2 6835 . . . . 5 (𝑓 = 𝑔 → (2nd𝑓) = (2nd𝑔))
76coeq2d 5812 . . . 4 (𝑓 = 𝑔 → (𝑡 ∘ (2nd𝑓)) = (𝑡 ∘ (2nd𝑔)))
85, 7opeq12d 4838 . . 3 (𝑓 = 𝑔 → ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩ = ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
94, 8cbvmpov 7455 . 2 (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩) = (𝑡𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
101, 9eqtri 2760 1 · = (𝑡𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cop 4587   × cxp 5623  ccom 5629  cfv 6493  cmpo 7362  1st c1st 7933  2nd c2nd 7934
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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-co 5634  df-iota 6449  df-fv 6501  df-oprab 7364  df-mpo 7365
This theorem is referenced by:  dvhvscaval  41396
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