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Theorem fovrn 5881
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 4541 . . 3  |-  ( ( A  e.  R  /\  B  e.  S )  -> 
<. A ,  B >.  e.  ( R  X.  S
) )
2 df-ov 5745 . . . 4  |-  ( A F B )  =  ( F `  <. A ,  B >. )
3 ffvelrn 5521 . . . 4  |-  ( ( F : ( R  X.  S ) --> C  /\  <. A ,  B >.  e.  ( R  X.  S ) )  -> 
( F `  <. A ,  B >. )  e.  C )
42, 3eqeltrid 2204 . . 3  |-  ( ( F : ( R  X.  S ) --> C  /\  <. A ,  B >.  e.  ( R  X.  S ) )  -> 
( A F B )  e.  C )
51, 4sylan2 284 . 2  |-  ( ( F : ( R  X.  S ) --> C  /\  ( A  e.  R  /\  B  e.  S ) )  -> 
( A F B )  e.  C )
653impb 1162 1  |-  ( ( F : ( R  X.  S ) --> C  /\  A  e.  R  /\  B  e.  S
)  ->  ( A F B )  e.  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 947    e. wcel 1465   <.cop 3500    X. cxp 4507   -->wf 5089   ` cfv 5093  (class class class)co 5742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-ov 5745
This theorem is referenced by:  fovrnda  5882  fovrnd  5883  ovmpoelrn  6073  mapxpen  6710  psmetcl  12422  xmetcl  12448  metcl  12449  blssm  12517
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