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Theorem fvovco 45136
Description: Value of the composition of an operator, with a given function. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypotheses
Ref Expression
fvovco.1 (𝜑𝐹:𝑋⟶(𝑉 × 𝑊))
fvovco.2 (𝜑𝑌𝑋)
Assertion
Ref Expression
fvovco (𝜑 → ((𝑂𝐹)‘𝑌) = ((1st ‘(𝐹𝑌))𝑂(2nd ‘(𝐹𝑌))))

Proof of Theorem fvovco
StepHypRef Expression
1 fvovco.1 . . . . 5 (𝜑𝐹:𝑋⟶(𝑉 × 𝑊))
2 fvovco.2 . . . . 5 (𝜑𝑌𝑋)
31, 2ffvelcdmd 7105 . . . 4 (𝜑 → (𝐹𝑌) ∈ (𝑉 × 𝑊))
4 1st2nd2 8052 . . . 4 ((𝐹𝑌) ∈ (𝑉 × 𝑊) → (𝐹𝑌) = ⟨(1st ‘(𝐹𝑌)), (2nd ‘(𝐹𝑌))⟩)
53, 4syl 17 . . 3 (𝜑 → (𝐹𝑌) = ⟨(1st ‘(𝐹𝑌)), (2nd ‘(𝐹𝑌))⟩)
65fveq2d 6911 . 2 (𝜑 → (𝑂‘(𝐹𝑌)) = (𝑂‘⟨(1st ‘(𝐹𝑌)), (2nd ‘(𝐹𝑌))⟩))
7 fvco3 7008 . . 3 ((𝐹:𝑋⟶(𝑉 × 𝑊) ∧ 𝑌𝑋) → ((𝑂𝐹)‘𝑌) = (𝑂‘(𝐹𝑌)))
81, 2, 7syl2anc 584 . 2 (𝜑 → ((𝑂𝐹)‘𝑌) = (𝑂‘(𝐹𝑌)))
9 df-ov 7434 . . 3 ((1st ‘(𝐹𝑌))𝑂(2nd ‘(𝐹𝑌))) = (𝑂‘⟨(1st ‘(𝐹𝑌)), (2nd ‘(𝐹𝑌))⟩)
109a1i 11 . 2 (𝜑 → ((1st ‘(𝐹𝑌))𝑂(2nd ‘(𝐹𝑌))) = (𝑂‘⟨(1st ‘(𝐹𝑌)), (2nd ‘(𝐹𝑌))⟩))
116, 8, 103eqtr4d 2785 1 (𝜑 → ((𝑂𝐹)‘𝑌) = ((1st ‘(𝐹𝑌))𝑂(2nd ‘(𝐹𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cop 4637   × cxp 5687  ccom 5693  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-1st 8013  df-2nd 8014
This theorem is referenced by:  cnmetcoval  45145  volicoff  45951  voliooicof  45952  hoissre  46500  hoiprodcl  46503  hoicvr  46504  hoicvrrex  46512  ovn0lem  46521  ovnhoilem1  46557  ovnhoilem2  46558  hoicoto2  46561  ovnlecvr2  46566  ovncvr2  46567  ovolval2lem  46599  ovolval5lem3  46610
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