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Theorem fovrnda 7308
Description: An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
Hypothesis
Ref Expression
fovrnd.1 (𝜑𝐹:(𝑅 × 𝑆)⟶𝐶)
Assertion
Ref Expression
fovrnda ((𝜑 ∧ (𝐴𝑅𝐵𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶)

Proof of Theorem fovrnda
StepHypRef Expression
1 fovrnd.1 . . 3 (𝜑𝐹:(𝑅 × 𝑆)⟶𝐶)
2 fovrn 7307 . . 3 ((𝐹:(𝑅 × 𝑆)⟶𝐶𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
31, 2syl3an1 1155 . 2 ((𝜑𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
433expb 1112 1 ((𝜑 ∧ (𝐴𝑅𝐵𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105   × cxp 5546  wf 6344  (class class class)co 7145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148
This theorem is referenced by:  eroprf  8384  yonedalem3  17518  yonedainv  17519  gass  18369  gsumxp2  19029  mamulid  20978  mamurid  20979  maducoeval2  21177  madutpos  21179  madugsum  21180  madurid  21181  isxmet2d  22864  prdsxmetlem  22905  rrxds  23923  ofrn  30314  fedgmullem2  30925  metideq  31032  sibfof  31497
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