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Theorem dvhvscacbv 35405
 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 6102 . . . 4 (𝑠 = 𝑡 → (𝑠‘(1st𝑓)) = (𝑡‘(1st𝑓)))
3 coeq1 5201 . . . 4 (𝑠 = 𝑡 → (𝑠 ∘ (2nd𝑓)) = (𝑡 ∘ (2nd𝑓)))
42, 3opeq12d 4348 . . 3 (𝑠 = 𝑡 → ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩ = ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩)
5 fveq2 6103 . . . . 5 (𝑓 = 𝑔 → (1st𝑓) = (1st𝑔))
65fveq2d 6107 . . . 4 (𝑓 = 𝑔 → (𝑡‘(1st𝑓)) = (𝑡‘(1st𝑔)))
7 fveq2 6103 . . . . 5 (𝑓 = 𝑔 → (2nd𝑓) = (2nd𝑔))
87coeq2d 5206 . . . 4 (𝑓 = 𝑔 → (𝑡 ∘ (2nd𝑓)) = (𝑡 ∘ (2nd𝑔)))
96, 8opeq12d 4348 . . 3 (𝑓 = 𝑔 → ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩ = ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
104, 9cbvmpt2v 6633 . 2 (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩) = (𝑡𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
111, 10eqtri 2632 1 · = (𝑡𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  ⟨cop 4131   × cxp 5036   ∘ ccom 5042  ‘cfv 5804   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-co 5047  df-iota 5768  df-fv 5812  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  dvhvscaval  35406
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