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Theorem opco1 7964
Description: Value of an operation precomposed with the projection on the first component. (Contributed by Mario Carneiro, 28-May-2014.) Generalize to closed form. (Revised by BJ, 27-Oct-2024.)
Hypotheses
Ref Expression
opco1.exa (𝜑𝐴𝑉)
opco1.exb (𝜑𝐵𝑊)
Assertion
Ref Expression
opco1 (𝜑 → (𝐴(𝐹 ∘ 1st )𝐵) = (𝐹𝐴))

Proof of Theorem opco1
StepHypRef Expression
1 df-ov 7278 . . 3 (𝐴(𝐹 ∘ 1st )𝐵) = ((𝐹 ∘ 1st )‘⟨𝐴, 𝐵⟩)
21a1i 11 . 2 (𝜑 → (𝐴(𝐹 ∘ 1st )𝐵) = ((𝐹 ∘ 1st )‘⟨𝐴, 𝐵⟩))
3 fo1st 7851 . . . 4 1st :V–onto→V
4 fof 6688 . . . 4 (1st :V–onto→V → 1st :V⟶V)
53, 4mp1i 13 . . 3 (𝜑 → 1st :V⟶V)
6 opex 5379 . . . 4 𝐴, 𝐵⟩ ∈ V
76a1i 11 . . 3 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ V)
85, 7fvco3d 6868 . 2 (𝜑 → ((𝐹 ∘ 1st )‘⟨𝐴, 𝐵⟩) = (𝐹‘(1st ‘⟨𝐴, 𝐵⟩)))
9 opco1.exa . . . 4 (𝜑𝐴𝑉)
10 opco1.exb . . . 4 (𝜑𝐵𝑊)
11 op1stg 7843 . . . 4 ((𝐴𝑉𝐵𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
129, 10, 11syl2anc 584 . . 3 (𝜑 → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
1312fveq2d 6778 . 2 (𝜑 → (𝐹‘(1st ‘⟨𝐴, 𝐵⟩)) = (𝐹𝐴))
142, 8, 133eqtrd 2782 1 (𝜑 → (𝐴(𝐹 ∘ 1st )𝐵) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  Vcvv 3432  cop 4567  ccom 5593  wf 6429  ontowfo 6431  cfv 6433  (class class class)co 7275  1st c1st 7829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-ov 7278  df-1st 7831
This theorem is referenced by:  opco1i  7966
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