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Theorem dvhvaddcbv 37164
Description: Change bound variables to isolate them later. (Contributed by NM, 3-Nov-2013.)
Hypothesis
Ref Expression
dvhvaddval.a + = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩)
Assertion
Ref Expression
dvhvaddcbv + = ( ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩)
Distinct variable groups:   𝑓,𝑔,,𝑖,𝐸   ,𝑓,𝑔,,𝑖   𝑇,𝑓,𝑔,,𝑖
Allowed substitution hints:   + (𝑓,𝑔,,𝑖)

Proof of Theorem dvhvaddcbv
StepHypRef Expression
1 dvhvaddval.a . 2 + = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩)
2 fveq2 6433 . . . . 5 (𝑓 = → (1st𝑓) = (1st))
32coeq1d 5516 . . . 4 (𝑓 = → ((1st𝑓) ∘ (1st𝑔)) = ((1st) ∘ (1st𝑔)))
4 fveq2 6433 . . . . 5 (𝑓 = → (2nd𝑓) = (2nd))
54oveq1d 6920 . . . 4 (𝑓 = → ((2nd𝑓) (2nd𝑔)) = ((2nd) (2nd𝑔)))
63, 5opeq12d 4631 . . 3 (𝑓 = → ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩ = ⟨((1st) ∘ (1st𝑔)), ((2nd) (2nd𝑔))⟩)
7 fveq2 6433 . . . . 5 (𝑔 = 𝑖 → (1st𝑔) = (1st𝑖))
87coeq2d 5517 . . . 4 (𝑔 = 𝑖 → ((1st) ∘ (1st𝑔)) = ((1st) ∘ (1st𝑖)))
9 fveq2 6433 . . . . 5 (𝑔 = 𝑖 → (2nd𝑔) = (2nd𝑖))
109oveq2d 6921 . . . 4 (𝑔 = 𝑖 → ((2nd) (2nd𝑔)) = ((2nd) (2nd𝑖)))
118, 10opeq12d 4631 . . 3 (𝑔 = 𝑖 → ⟨((1st) ∘ (1st𝑔)), ((2nd) (2nd𝑔))⟩ = ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩)
126, 11cbvmpt2v 6995 . 2 (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩) = ( ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩)
131, 12eqtri 2849 1 + = ( ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1658  cop 4403   × cxp 5340  ccom 5346  cfv 6123  (class class class)co 6905  cmpt2 6907  1st c1st 7426  2nd c2nd 7427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-co 5351  df-iota 6086  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910
This theorem is referenced by:  dvhvaddval  37165
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