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Theorem fovrn 6228
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 4866 . . 3  |-  ( ( A  e.  R  /\  B  e.  S )  -> 
<. A ,  B >.  e.  ( R  X.  S
) )
2 df-ov 6089 . . . 4  |-  ( A F B )  =  ( F `  <. A ,  B >. )
3 ffvelrn 5836 . . . 4  |-  ( ( F : ( R  X.  S ) --> C  /\  <. A ,  B >.  e.  ( R  X.  S ) )  -> 
( F `  <. A ,  B >. )  e.  C )
42, 3syl5eqel 2522 . . 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 1183 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 965    e. wcel 1756   <.cop 3878    X. cxp 4833   -->wf 5409   ` cfv 5413  (class class class)co 6086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6089
This theorem is referenced by:  fovrnda  6229  fovrnd  6230  curry1f  6661  curry2f  6663  mapxpen  7469  axdc4lem  8616  axdc4uzlem  11796  imasmnd2  15450  grpsubcl  15597  imasgrp2  15661  imasrng  16701  tsmsxplem1  19707  psmetcl  19863  xmetcl  19886  metcl  19887  blssm  19973  mbfi1fseqlem3  21175  mbfi1fseqlem4  21176  mbfi1fseqlem5  21177  grpocl  23655  grpodivcl  23702  clmgm  23776  rngocl  23837  vccl  23896  nvmcl  23995  cvmliftphtlem  27175  isbnd3  28654  isdrngo2  28735
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