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Theorem dvhvaddcbv 36797
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 6304 . . . . 5 (𝑓 = → (1st𝑓) = (1st))
32coeq1d 5391 . . . 4 (𝑓 = → ((1st𝑓) ∘ (1st𝑔)) = ((1st) ∘ (1st𝑔)))
4 fveq2 6304 . . . . 5 (𝑓 = → (2nd𝑓) = (2nd))
54oveq1d 6780 . . . 4 (𝑓 = → ((2nd𝑓) (2nd𝑔)) = ((2nd) (2nd𝑔)))
63, 5opeq12d 4517 . . 3 (𝑓 = → ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩ = ⟨((1st) ∘ (1st𝑔)), ((2nd) (2nd𝑔))⟩)
7 fveq2 6304 . . . . 5 (𝑔 = 𝑖 → (1st𝑔) = (1st𝑖))
87coeq2d 5392 . . . 4 (𝑔 = 𝑖 → ((1st) ∘ (1st𝑔)) = ((1st) ∘ (1st𝑖)))
9 fveq2 6304 . . . . 5 (𝑔 = 𝑖 → (2nd𝑔) = (2nd𝑖))
109oveq2d 6781 . . . 4 (𝑔 = 𝑖 → ((2nd) (2nd𝑔)) = ((2nd) (2nd𝑖)))
118, 10opeq12d 4517 . . 3 (𝑔 = 𝑖 → ⟨((1st) ∘ (1st𝑔)), ((2nd) (2nd𝑔))⟩ = ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩)
126, 11cbvmpt2v 6852 . 2 (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩) = ( ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩)
131, 12eqtri 2746 1 + = ( ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1596  cop 4291   × cxp 5216  ccom 5222  cfv 6001  (class class class)co 6765  cmpt2 6767  1st c1st 7283  2nd c2nd 7284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-co 5227  df-iota 5964  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770
This theorem is referenced by:  dvhvaddval  36798
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