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Theorem fvovco 40135
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 6586 . . . 4 (𝜑 → (𝐹𝑌) ∈ (𝑉 × 𝑊))
4 1st2nd2 7440 . . . 4 ((𝐹𝑌) ∈ (𝑉 × 𝑊) → (𝐹𝑌) = ⟨(1st ‘(𝐹𝑌)), (2nd ‘(𝐹𝑌))⟩)
53, 4syl 17 . . 3 (𝜑 → (𝐹𝑌) = ⟨(1st ‘(𝐹𝑌)), (2nd ‘(𝐹𝑌))⟩)
65fveq2d 6415 . 2 (𝜑 → (𝑂‘(𝐹𝑌)) = (𝑂‘⟨(1st ‘(𝐹𝑌)), (2nd ‘(𝐹𝑌))⟩))
7 fvco3 6500 . . 3 ((𝐹:𝑋⟶(𝑉 × 𝑊) ∧ 𝑌𝑋) → ((𝑂𝐹)‘𝑌) = (𝑂‘(𝐹𝑌)))
81, 2, 7syl2anc 580 . 2 (𝜑 → ((𝑂𝐹)‘𝑌) = (𝑂‘(𝐹𝑌)))
9 df-ov 6881 . . 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 2843 1 (𝜑 → ((𝑂𝐹)‘𝑌) = ((1st ‘(𝐹𝑌))𝑂(2nd ‘(𝐹𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  cop 4374   × cxp 5310  ccom 5316  wf 6097  cfv 6101  (class class class)co 6878  1st c1st 7399  2nd c2nd 7400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-1st 7401  df-2nd 7402
This theorem is referenced by:  cnmetcoval  40146  volicoff  40955  voliooicof  40956  hoissre  41504  hoiprodcl  41507  hoicvr  41508  hoicvrrex  41516  ovn0lem  41525  ovnhoilem1  41561  ovnhoilem2  41562  hoicoto2  41565  ovnlecvr2  41570  ovncvr2  41571  ovolval2lem  41603  ovolval5lem3  41614
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