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Theorem dvhvaddcbv 40562
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 6897 . . . . 5 (𝑓 = → (1st𝑓) = (1st))
32coeq1d 5864 . . . 4 (𝑓 = → ((1st𝑓) ∘ (1st𝑔)) = ((1st) ∘ (1st𝑔)))
4 fveq2 6897 . . . . 5 (𝑓 = → (2nd𝑓) = (2nd))
54oveq1d 7435 . . . 4 (𝑓 = → ((2nd𝑓) (2nd𝑔)) = ((2nd) (2nd𝑔)))
63, 5opeq12d 4882 . . 3 (𝑓 = → ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩ = ⟨((1st) ∘ (1st𝑔)), ((2nd) (2nd𝑔))⟩)
7 fveq2 6897 . . . . 5 (𝑔 = 𝑖 → (1st𝑔) = (1st𝑖))
87coeq2d 5865 . . . 4 (𝑔 = 𝑖 → ((1st) ∘ (1st𝑔)) = ((1st) ∘ (1st𝑖)))
9 fveq2 6897 . . . . 5 (𝑔 = 𝑖 → (2nd𝑔) = (2nd𝑖))
109oveq2d 7436 . . . 4 (𝑔 = 𝑖 → ((2nd) (2nd𝑔)) = ((2nd) (2nd𝑖)))
118, 10opeq12d 4882 . . 3 (𝑔 = 𝑖 → ⟨((1st) ∘ (1st𝑔)), ((2nd) (2nd𝑔))⟩ = ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩)
126, 11cbvmpov 7515 . 2 (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩) = ( ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩)
131, 12eqtri 2756 1 + = ( ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  cop 4635   × cxp 5676  ccom 5682  cfv 6548  (class class class)co 7420  cmpo 7422  1st c1st 7991  2nd c2nd 7992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-co 5687  df-iota 6500  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425
This theorem is referenced by:  dvhvaddval  40563
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