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Theorem ffvelrnb 25131
Description: A function's value belongs to its codomain. (Contributed by FL, 14-Sep-2013.)
Assertion
Ref Expression
ffvelrnb  |-  ( ( A  e.  D  /\  B  e.  E )  ->  ( ( F  e.  ( B  ^m  A
)  /\  C  e.  A )  ->  ( F `  C )  e.  B ) )

Proof of Theorem ffvelrnb
StepHypRef Expression
1 elmapg 6785 . . . 4  |-  ( ( B  e.  E  /\  A  e.  D )  ->  ( F  e.  ( B  ^m  A )  <-> 
F : A --> B ) )
21ancoms 439 . . 3  |-  ( ( A  e.  D  /\  B  e.  E )  ->  ( F  e.  ( B  ^m  A )  <-> 
F : A --> B ) )
3 ffvelrn 5663 . . . 4  |-  ( ( F : A --> B  /\  C  e.  A )  ->  ( F `  C
)  e.  B )
43ex 423 . . 3  |-  ( F : A --> B  -> 
( C  e.  A  ->  ( F `  C
)  e.  B ) )
52, 4syl6bi 219 . 2  |-  ( ( A  e.  D  /\  B  e.  E )  ->  ( F  e.  ( B  ^m  A )  ->  ( C  e.  A  ->  ( F `  C )  e.  B
) ) )
65imp3a 420 1  |-  ( ( A  e.  D  /\  B  e.  E )  ->  ( ( F  e.  ( B  ^m  A
)  /\  C  e.  A )  ->  ( F `  C )  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772
This theorem is referenced by:  addassv  25656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774
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