MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fovrn Structured version   Unicode version

Theorem fovrn 6430
Description: An operation's value belongs to its codomain. (Contributed by NM, 27-Aug-2006.)
Assertion
Ref Expression
fovrn  |-  ( ( F : ( R  X.  S ) --> C  /\  A  e.  R  /\  B  e.  S
)  ->  ( A F B )  e.  C
)

Proof of Theorem fovrn
StepHypRef Expression
1 opelxpi 5021 . . 3  |-  ( ( A  e.  R  /\  B  e.  S )  -> 
<. A ,  B >.  e.  ( R  X.  S
) )
2 df-ov 6284 . . . 4  |-  ( A F B )  =  ( F `  <. A ,  B >. )
3 ffvelrn 6014 . . . 4  |-  ( ( F : ( R  X.  S ) --> C  /\  <. A ,  B >.  e.  ( R  X.  S ) )  -> 
( F `  <. A ,  B >. )  e.  C )
42, 3syl5eqel 2535 . . 3  |-  ( ( F : ( R  X.  S ) --> C  /\  <. A ,  B >.  e.  ( R  X.  S ) )  -> 
( A F B )  e.  C )
51, 4sylan2 474 . 2  |-  ( ( F : ( R  X.  S ) --> C  /\  ( A  e.  R  /\  B  e.  S ) )  -> 
( A F B )  e.  C )
653impb 1193 1  |-  ( ( F : ( R  X.  S ) --> C  /\  A  e.  R  /\  B  e.  S
)  ->  ( A F B )  e.  C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    e. wcel 1804   <.cop 4020    X. cxp 4987   -->wf 5574   ` cfv 5578  (class class class)co 6281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586  df-ov 6284
This theorem is referenced by:  fovrnda  6431  fovrnd  6432  ovmpt2elrn  6856  curry1f  6879  curry2f  6881  mapxpen  7685  axdc4lem  8838  axdc4uzlem  12074  imasmnd2  15936  grpsubcl  16097  imasgrp2  16164  imasring  17247  tsmsxplem1  20633  psmetcl  20789  xmetcl  20812  metcl  20813  blssm  20899  mbfi1fseqlem3  22102  mbfi1fseqlem4  22103  mbfi1fseqlem5  22104  grpocl  25180  grpodivcl  25227  clmgmOLD  25301  rngocl  25362  vccl  25421  nvmcl  25520  cvmliftphtlem  28740  isbnd3  30256  isdrngo2  30337
  Copyright terms: Public domain W3C validator