MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fovcdmda Structured version   Visualization version   GIF version

Theorem fovcdmda 7574
Description: An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
Hypothesis
Ref Expression
fovcdmd.1 (𝜑𝐹:(𝑅 × 𝑆)⟶𝐶)
Assertion
Ref Expression
fovcdmda ((𝜑 ∧ (𝐴𝑅𝐵𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶)

Proof of Theorem fovcdmda
StepHypRef Expression
1 fovcdmd.1 . . 3 (𝜑𝐹:(𝑅 × 𝑆)⟶𝐶)
2 fovcdm 7573 . . 3 ((𝐹:(𝑅 × 𝑆)⟶𝐶𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
31, 2syl3an1 1163 . 2 ((𝜑𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
433expb 1120 1 ((𝜑 ∧ (𝐴𝑅𝐵𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106   × cxp 5673  wf 6536  (class class class)co 7405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408
This theorem is referenced by:  eroprf  8805  yonedalem3  18229  yonedainv  18230  gass  19159  gsumxp2  19842  mamulid  21934  mamurid  21935  maducoeval2  22133  madutpos  22135  madugsum  22136  madurid  22137  isxmet2d  23824  prdsxmetlem  23865  rrxds  24901  ofrn  31851  fedgmullem2  32703  metideq  32861  sibfof  33327  ofoacl  42092  naddcnfcl  42100
  Copyright terms: Public domain W3C validator