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Theorem dvhvscacbv 38709
 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 6662 . . . 4 (𝑠 = 𝑡 → (𝑠‘(1st𝑓)) = (𝑡‘(1st𝑓)))
3 coeq1 5703 . . . 4 (𝑠 = 𝑡 → (𝑠 ∘ (2nd𝑓)) = (𝑡 ∘ (2nd𝑓)))
42, 3opeq12d 4774 . . 3 (𝑠 = 𝑡 → ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩ = ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩)
5 2fveq3 6668 . . . 4 (𝑓 = 𝑔 → (𝑡‘(1st𝑓)) = (𝑡‘(1st𝑔)))
6 fveq2 6663 . . . . 5 (𝑓 = 𝑔 → (2nd𝑓) = (2nd𝑔))
76coeq2d 5708 . . . 4 (𝑓 = 𝑔 → (𝑡 ∘ (2nd𝑓)) = (𝑡 ∘ (2nd𝑔)))
85, 7opeq12d 4774 . . 3 (𝑓 = 𝑔 → ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩ = ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
94, 8cbvmpov 7249 . 2 (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩) = (𝑡𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
101, 9eqtri 2781 1 · = (𝑡𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  ⟨cop 4531   × cxp 5526   ∘ ccom 5532  ‘cfv 6340   ∈ cmpo 7158  1st c1st 7697  2nd c2nd 7698 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-co 5537  df-iota 6299  df-fv 6348  df-oprab 7160  df-mpo 7161 This theorem is referenced by:  dvhvscaval  38710
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