ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fovcdmd GIF version

Theorem fovcdmd 6018
Description: An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
Hypotheses
Ref Expression
fovcdmd.1 (𝜑𝐹:(𝑅 × 𝑆)⟶𝐶)
fovcdmd.2 (𝜑𝐴𝑅)
fovcdmd.3 (𝜑𝐵𝑆)
Assertion
Ref Expression
fovcdmd (𝜑 → (𝐴𝐹𝐵) ∈ 𝐶)

Proof of Theorem fovcdmd
StepHypRef Expression
1 fovcdmd.1 . 2 (𝜑𝐹:(𝑅 × 𝑆)⟶𝐶)
2 fovcdmd.2 . 2 (𝜑𝐴𝑅)
3 fovcdmd.3 . 2 (𝜑𝐵𝑆)
4 fovcdm 6016 . 2 ((𝐹:(𝑅 × 𝑆)⟶𝐶𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
51, 2, 3, 4syl3anc 1238 1 (𝜑 → (𝐴𝐹𝐵) ∈ 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2148   × cxp 4624  wf 5212  (class class class)co 5874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-ov 5877
This theorem is referenced by:  eroveu  6625  isxmet2d  13741  ismet2  13747  comet  13892  bdmetval  13893  txmetcnp  13911  limccnp2lem  14038  limccnp2cntop  14039
  Copyright terms: Public domain W3C validator