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Theorem resfunexgALT 5944
Description: The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. This version has a shorter proof than resfunexg 5943 but requires ax-pow 4364. (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 5155 . . . 4  |-  ( B  e.  C  ->  dom  ( A  |`  B )  e.  _V )
21adantl 453 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  dom  ( A  |`  B )  e.  _V )
3 df-ima 4877 . . . 4  |-  ( A
" B )  =  ran  ( A  |`  B )
4 funimaexg 5516 . . . 4  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A " B )  e. 
_V )
53, 4syl5eqelr 2515 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  ran  ( A  |`  B )  e.  _V )
62, 5jca 519 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( dom  ( A  |`  B )  e.  _V  /\  ran  ( A  |`  B )  e.  _V ) )
7 xpexg 4975 . 2  |-  ( ( dom  ( A  |`  B )  e.  _V  /\ 
ran  ( A  |`  B )  e.  _V )  ->  ( dom  ( A  |`  B )  X. 
ran  ( A  |`  B ) )  e. 
_V )
8 relres 5160 . . . 4  |-  Rel  ( A  |`  B )
9 relssdmrn 5376 . . . 4  |-  ( Rel  ( A  |`  B )  ->  ( A  |`  B )  C_  ( dom  ( A  |`  B )  X.  ran  ( A  |`  B ) ) )
108, 9ax-mp 8 . . 3  |-  ( A  |`  B )  C_  ( dom  ( A  |`  B )  X.  ran  ( A  |`  B ) )
11 ssexg 4336 . . 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 652 . 2  |-  ( ( dom  ( A  |`  B )  X.  ran  ( A  |`  B ) )  e.  _V  ->  ( A  |`  B )  e.  _V )
136, 7, 123syl 19 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   _Vcvv 2943    C_ wss 3307    X. cxp 4862   dom cdm 4864   ran crn 4865    |` cres 4866   "cima 4867   Rel wrel 4869   Fun wfun 5434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-fun 5442
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