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

Theorem fovcdm 7603
Description: An operation's value belongs to its codomain. (Contributed by NM, 27-Aug-2006.)
Assertion
Ref Expression
fovcdm ((𝐹:(𝑅 × 𝑆)⟶𝐶𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)

Proof of Theorem fovcdm
StepHypRef Expression
1 opelxpi 5722 . . 3 ((𝐴𝑅𝐵𝑆) → ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆))
2 df-ov 7434 . . . 4 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
3 ffvelcdm 7101 . . . 4 ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆)) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ 𝐶)
42, 3eqeltrid 2845 . . 3 ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶)
51, 4sylan2 593 . 2 ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ (𝐴𝑅𝐵𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶)
653impb 1115 1 ((𝐹:(𝑅 × 𝑆)⟶𝐶𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087  wcel 2108  cop 4632   × cxp 5683  wf 6557  cfv 6561  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434
This theorem is referenced by:  fovcdmda  7604  fovcdmd  7605  ovmpoelrn  8097  curry1f  8131  curry2f  8133  mapxpen  9183  axdc4lem  10495  axdc4uzlem  14024  imasmnd2  18787  grpsubcl  19038  imasgrp2  19073  imasring  20327  tsmsxplem1  24161  psmetcl  24317  xmetcl  24341  metcl  24342  blssm  24428  mbfi1fseqlem3  25752  mbfi1fseqlem4  25753  mbfi1fseqlem5  25754  grpocl  30519  grpodivcl  30558  vccl  30582  nvmcl  30665  cvmliftphtlem  35322  matunitlindflem1  37623  isbnd3  37791  clmgmOLD  37858  rngocl  37908  isdrngo2  37965
  Copyright terms: Public domain W3C validator