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Theorem fovrn 6342
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 4978 . . 3  |-  ( ( A  e.  R  /\  B  e.  S )  -> 
<. A ,  B >.  e.  ( R  X.  S
) )
2 df-ov 6202 . . . 4  |-  ( A F B )  =  ( F `  <. A ,  B >. )
3 ffvelrn 5949 . . . 4  |-  ( ( F : ( R  X.  S ) --> C  /\  <. A ,  B >.  e.  ( R  X.  S ) )  -> 
( F `  <. A ,  B >. )  e.  C )
42, 3syl5eqel 2546 . . 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 1184 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 1758   <.cop 3990    X. cxp 4945   -->wf 5521   ` cfv 5525  (class class class)co 6199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-ov 6202
This theorem is referenced by:  fovrnda  6343  fovrnd  6344  curry1f  6775  curry2f  6777  mapxpen  7586  axdc4lem  8734  axdc4uzlem  11920  imasmnd2  15576  grpsubcl  15724  imasgrp2  15788  imasrng  16833  tsmsxplem1  19858  psmetcl  20014  xmetcl  20037  metcl  20038  blssm  20124  mbfi1fseqlem3  21327  mbfi1fseqlem4  21328  mbfi1fseqlem5  21329  grpocl  23838  grpodivcl  23885  clmgm  23959  rngocl  24020  vccl  24079  nvmcl  24178  cvmliftphtlem  27349  isbnd3  28830  isdrngo2  28911
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