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Theorem opco2 8149
Description: Value of an operation precomposed with the projection on the second component. (Contributed by BJ, 27-Oct-2024.)
Hypotheses
Ref Expression
opco1.exa (𝜑𝐴𝑉)
opco1.exb (𝜑𝐵𝑊)
Assertion
Ref Expression
opco2 (𝜑 → (𝐴(𝐹 ∘ 2nd )𝐵) = (𝐹𝐵))

Proof of Theorem opco2
StepHypRef Expression
1 df-ov 7434 . . 3 (𝐴(𝐹 ∘ 2nd )𝐵) = ((𝐹 ∘ 2nd )‘⟨𝐴, 𝐵⟩)
21a1i 11 . 2 (𝜑 → (𝐴(𝐹 ∘ 2nd )𝐵) = ((𝐹 ∘ 2nd )‘⟨𝐴, 𝐵⟩))
3 fo2nd 8035 . . . 4 2nd :V–onto→V
4 fof 6820 . . . 4 (2nd :V–onto→V → 2nd :V⟶V)
53, 4mp1i 13 . . 3 (𝜑 → 2nd :V⟶V)
6 opex 5469 . . . 4 𝐴, 𝐵⟩ ∈ V
76a1i 11 . . 3 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ V)
85, 7fvco3d 7009 . 2 (𝜑 → ((𝐹 ∘ 2nd )‘⟨𝐴, 𝐵⟩) = (𝐹‘(2nd ‘⟨𝐴, 𝐵⟩)))
9 opco1.exa . . . 4 (𝜑𝐴𝑉)
10 opco1.exb . . . 4 (𝜑𝐵𝑊)
11 op2ndg 8027 . . . 4 ((𝐴𝑉𝐵𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
129, 10, 11syl2anc 584 . . 3 (𝜑 → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
1312fveq2d 6910 . 2 (𝜑 → (𝐹‘(2nd ‘⟨𝐴, 𝐵⟩)) = (𝐹𝐵))
142, 8, 133eqtrd 2781 1 (𝜑 → (𝐴(𝐹 ∘ 2nd )𝐵) = (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632  ccom 5689  wf 6557  ontowfo 6559  cfv 6561  (class class class)co 7431  2nd c2nd 8013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-ov 7434  df-2nd 8015
This theorem is referenced by:  dfwrecsOLD  8338  wfr2a  8374  dfrecs3  8412
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