Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fvovco Structured version   Visualization version   GIF version

Theorem fvovco 43892
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 7088 . . . 4 (𝜑 → (𝐹𝑌) ∈ (𝑉 × 𝑊))
4 1st2nd2 8014 . . . 4 ((𝐹𝑌) ∈ (𝑉 × 𝑊) → (𝐹𝑌) = ⟨(1st ‘(𝐹𝑌)), (2nd ‘(𝐹𝑌))⟩)
53, 4syl 17 . . 3 (𝜑 → (𝐹𝑌) = ⟨(1st ‘(𝐹𝑌)), (2nd ‘(𝐹𝑌))⟩)
65fveq2d 6896 . 2 (𝜑 → (𝑂‘(𝐹𝑌)) = (𝑂‘⟨(1st ‘(𝐹𝑌)), (2nd ‘(𝐹𝑌))⟩))
7 fvco3 6991 . . 3 ((𝐹:𝑋⟶(𝑉 × 𝑊) ∧ 𝑌𝑋) → ((𝑂𝐹)‘𝑌) = (𝑂‘(𝐹𝑌)))
81, 2, 7syl2anc 585 . 2 (𝜑 → ((𝑂𝐹)‘𝑌) = (𝑂‘(𝐹𝑌)))
9 df-ov 7412 . . 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 2783 1 (𝜑 → ((𝑂𝐹)‘𝑌) = ((1st ‘(𝐹𝑌))𝑂(2nd ‘(𝐹𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cop 4635   × cxp 5675  ccom 5681  wf 6540  cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-1st 7975  df-2nd 7976
This theorem is referenced by:  cnmetcoval  43901  volicoff  44711  voliooicof  44712  hoissre  45260  hoiprodcl  45263  hoicvr  45264  hoicvrrex  45272  ovn0lem  45281  ovnhoilem1  45317  ovnhoilem2  45318  hoicoto2  45321  ovnlecvr2  45326  ovncvr2  45327  ovolval2lem  45359  ovolval5lem3  45370
  Copyright terms: Public domain W3C validator