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Theorem dvhvscacbv 37118
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 6411 . . . 4 (𝑠 = 𝑡 → (𝑠‘(1st𝑓)) = (𝑡‘(1st𝑓)))
3 coeq1 5484 . . . 4 (𝑠 = 𝑡 → (𝑠 ∘ (2nd𝑓)) = (𝑡 ∘ (2nd𝑓)))
42, 3opeq12d 4602 . . 3 (𝑠 = 𝑡 → ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩ = ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩)
5 2fveq3 6417 . . . 4 (𝑓 = 𝑔 → (𝑡‘(1st𝑓)) = (𝑡‘(1st𝑔)))
6 fveq2 6412 . . . . 5 (𝑓 = 𝑔 → (2nd𝑓) = (2nd𝑔))
76coeq2d 5489 . . . 4 (𝑓 = 𝑔 → (𝑡 ∘ (2nd𝑓)) = (𝑡 ∘ (2nd𝑔)))
85, 7opeq12d 4602 . . 3 (𝑓 = 𝑔 → ⟨(𝑡‘(1st𝑓)), (𝑡 ∘ (2nd𝑓))⟩ = ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
94, 8cbvmpt2v 6970 . 2 (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩) = (𝑡𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
101, 9eqtri 2822 1 · = (𝑡𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  cop 4375   × cxp 5311  ccom 5317  cfv 6102  cmpt2 6881  1st c1st 7400  2nd c2nd 7401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-rex 3096  df-rab 3099  df-v 3388  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-co 5322  df-iota 6065  df-fv 6110  df-oprab 6883  df-mpt2 6884
This theorem is referenced by:  dvhvscaval  37119
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