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Theorem fvovco 42732
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, 2ffvelrnd 6962 . . . 4 (𝜑 → (𝐹𝑌) ∈ (𝑉 × 𝑊))
4 1st2nd2 7870 . . . 4 ((𝐹𝑌) ∈ (𝑉 × 𝑊) → (𝐹𝑌) = ⟨(1st ‘(𝐹𝑌)), (2nd ‘(𝐹𝑌))⟩)
53, 4syl 17 . . 3 (𝜑 → (𝐹𝑌) = ⟨(1st ‘(𝐹𝑌)), (2nd ‘(𝐹𝑌))⟩)
65fveq2d 6778 . 2 (𝜑 → (𝑂‘(𝐹𝑌)) = (𝑂‘⟨(1st ‘(𝐹𝑌)), (2nd ‘(𝐹𝑌))⟩))
7 fvco3 6867 . . 3 ((𝐹:𝑋⟶(𝑉 × 𝑊) ∧ 𝑌𝑋) → ((𝑂𝐹)‘𝑌) = (𝑂‘(𝐹𝑌)))
81, 2, 7syl2anc 584 . 2 (𝜑 → ((𝑂𝐹)‘𝑌) = (𝑂‘(𝐹𝑌)))
9 df-ov 7278 . . 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 2788 1 (𝜑 → ((𝑂𝐹)‘𝑌) = ((1st ‘(𝐹𝑌))𝑂(2nd ‘(𝐹𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  cop 4567   × cxp 5587  ccom 5593  wf 6429  cfv 6433  (class class class)co 7275  1st c1st 7829  2nd c2nd 7830
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-fv 6441  df-ov 7278  df-1st 7831  df-2nd 7832
This theorem is referenced by:  cnmetcoval  42742  volicoff  43536  voliooicof  43537  hoissre  44082  hoiprodcl  44085  hoicvr  44086  hoicvrrex  44094  ovn0lem  44103  ovnhoilem1  44139  ovnhoilem2  44140  hoicoto2  44143  ovnlecvr2  44148  ovncvr2  44149  ovolval2lem  44181  ovolval5lem3  44192
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