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 42254
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 6856 . . . 4 (𝜑 → (𝐹𝑌) ∈ (𝑉 × 𝑊))
4 1st2nd2 7746 . . . 4 ((𝐹𝑌) ∈ (𝑉 × 𝑊) → (𝐹𝑌) = ⟨(1st ‘(𝐹𝑌)), (2nd ‘(𝐹𝑌))⟩)
53, 4syl 17 . . 3 (𝜑 → (𝐹𝑌) = ⟨(1st ‘(𝐹𝑌)), (2nd ‘(𝐹𝑌))⟩)
65fveq2d 6672 . 2 (𝜑 → (𝑂‘(𝐹𝑌)) = (𝑂‘⟨(1st ‘(𝐹𝑌)), (2nd ‘(𝐹𝑌))⟩))
7 fvco3 6761 . . 3 ((𝐹:𝑋⟶(𝑉 × 𝑊) ∧ 𝑌𝑋) → ((𝑂𝐹)‘𝑌) = (𝑂‘(𝐹𝑌)))
81, 2, 7syl2anc 587 . 2 (𝜑 → ((𝑂𝐹)‘𝑌) = (𝑂‘(𝐹𝑌)))
9 df-ov 7167 . . 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 2113  cop 4519   × cxp 5517  ccom 5523  wf 6329  cfv 6333  (class class class)co 7164  1st c1st 7705  2nd c2nd 7706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-fv 6341  df-ov 7167  df-1st 7707  df-2nd 7708
This theorem is referenced by:  cnmetcoval  42264  volicoff  43062  voliooicof  43063  hoissre  43608  hoiprodcl  43611  hoicvr  43612  hoicvrrex  43620  ovn0lem  43629  ovnhoilem1  43665  ovnhoilem2  43666  hoicoto2  43669  ovnlecvr2  43674  ovncvr2  43675  ovolval2lem  43707  ovolval5lem3  43718
  Copyright terms: Public domain W3C validator