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Theorem resfunexgALT 6753
Description: Alternate proof of resfunexg 6128, shorter but requiring ax-pow 4580 and ax-un 6580. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
resfunexgALT  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  e. 
_V )

Proof of Theorem resfunexgALT
StepHypRef Expression
1 dmresexg 5126 . . . 4  |-  ( B  e.  C  ->  dom  ( A  |`  B )  e.  _V )
21adantl 468 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  dom  ( A  |`  B )  e.  _V )
3 df-ima 4846 . . . 4  |-  ( A
" B )  =  ran  ( A  |`  B )
4 funimaexg 5658 . . . 4  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A " B )  e. 
_V )
53, 4syl5eqelr 2533 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  ran  ( A  |`  B )  e.  _V )
62, 5jca 535 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( dom  ( A  |`  B )  e.  _V  /\  ran  ( A  |`  B )  e.  _V ) )
7 xpexg 6590 . 2  |-  ( ( dom  ( A  |`  B )  e.  _V  /\ 
ran  ( A  |`  B )  e.  _V )  ->  ( dom  ( A  |`  B )  X. 
ran  ( A  |`  B ) )  e. 
_V )
8 relres 5131 . . . 4  |-  Rel  ( A  |`  B )
9 relssdmrn 5355 . . . 4  |-  ( Rel  ( A  |`  B )  ->  ( A  |`  B )  C_  ( dom  ( A  |`  B )  X.  ran  ( A  |`  B ) ) )
108, 9ax-mp 5 . . 3  |-  ( A  |`  B )  C_  ( dom  ( A  |`  B )  X.  ran  ( A  |`  B ) )
11 ssexg 4548 . . 3  |-  ( ( ( A  |`  B ) 
C_  ( dom  ( A  |`  B )  X. 
ran  ( A  |`  B ) )  /\  ( dom  ( A  |`  B )  X.  ran  ( A  |`  B ) )  e.  _V )  ->  ( A  |`  B )  e.  _V )
1210, 11mpan 675 . 2  |-  ( ( dom  ( A  |`  B )  X.  ran  ( A  |`  B ) )  e.  _V  ->  ( A  |`  B )  e.  _V )
136, 7, 123syl 18 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  e. 
_V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    e. wcel 1886   _Vcvv 3044    C_ wss 3403    X. cxp 4831   dom cdm 4833   ran crn 4834    |` cres 4835   "cima 4836   Rel wrel 4838   Fun wfun 5575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-fun 5583
This theorem is referenced by: (None)
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