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Theorem fovrnda 7313
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 7312 . . 3 ((𝐹:(𝑅 × 𝑆)⟶𝐶𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
31, 2syl3an1 1157 . 2 ((𝜑𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
433expb 1114 1 ((𝜑 ∧ (𝐴𝑅𝐵𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2107   × cxp 5552  wf 6350  (class class class)co 7150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-ov 7153
This theorem is referenced by:  eroprf  8390  yonedalem3  17525  yonedainv  17526  gass  18376  gsumxp2  19036  mamulid  20985  mamurid  20986  maducoeval2  21184  madutpos  21186  madugsum  21187  madurid  21188  isxmet2d  22871  prdsxmetlem  22912  rrxds  23930  ofrn  30320  fedgmullem2  30931  metideq  31038  sibfof  31503
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