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Theorem funimaexg 5202
Description: Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 10-Sep-2006.)
Assertion
Ref Expression
funimaexg  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A " B )  e. 
_V )

Proof of Theorem funimaexg
StepHypRef Expression
1 simpl 108 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  Fun  A )
2 funrel 5135 . . 3  |-  ( Fun 
A  ->  Rel  A )
3 resres 4826 . . . . . . 7  |-  ( ( A  |`  dom  A )  |`  B )  =  ( A  |`  ( dom  A  i^i  B ) )
4 incom 3263 . . . . . . . 8  |-  ( B  i^i  dom  A )  =  ( dom  A  i^i  B )
54reseq2i 4811 . . . . . . 7  |-  ( A  |`  ( B  i^i  dom  A ) )  =  ( A  |`  ( dom  A  i^i  B ) )
63, 5eqtr4i 2161 . . . . . 6  |-  ( ( A  |`  dom  A )  |`  B )  =  ( A  |`  ( B  i^i  dom  A ) )
7 resdm 4853 . . . . . . 7  |-  ( Rel 
A  ->  ( A  |` 
dom  A )  =  A )
87reseq1d 4813 . . . . . 6  |-  ( Rel 
A  ->  ( ( A  |`  dom  A )  |`  B )  =  ( A  |`  B )
)
96, 8syl5eqr 2184 . . . . 5  |-  ( Rel 
A  ->  ( A  |`  ( B  i^i  dom  A ) )  =  ( A  |`  B )
)
109rneqd 4763 . . . 4  |-  ( Rel 
A  ->  ran  ( A  |`  ( B  i^i  dom  A ) )  =  ran  ( A  |`  B ) )
11 df-ima 4547 . . . 4  |-  ( A
" ( B  i^i  dom 
A ) )  =  ran  ( A  |`  ( B  i^i  dom  A
) )
12 df-ima 4547 . . . 4  |-  ( A
" B )  =  ran  ( A  |`  B )
1310, 11, 123eqtr4g 2195 . . 3  |-  ( Rel 
A  ->  ( A " ( B  i^i  dom  A ) )  =  ( A " B ) )
141, 2, 133syl 17 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A " ( B  i^i  dom 
A ) )  =  ( A " B
) )
15 inex1g 4059 . . 3  |-  ( B  e.  C  ->  ( B  i^i  dom  A )  e.  _V )
16 inss2 3292 . . . 4  |-  ( B  i^i  dom  A )  C_ 
dom  A
17 funimaexglem 5201 . . . 4  |-  ( ( Fun  A  /\  ( B  i^i  dom  A )  e.  _V  /\  ( B  i^i  dom  A )  C_ 
dom  A )  -> 
( A " ( B  i^i  dom  A )
)  e.  _V )
1816, 17mp3an3 1304 . . 3  |-  ( ( Fun  A  /\  ( B  i^i  dom  A )  e.  _V )  ->  ( A " ( B  i^i  dom 
A ) )  e. 
_V )
1915, 18sylan2 284 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A " ( B  i^i  dom 
A ) )  e. 
_V )
2014, 19eqeltrrd 2215 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A " B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   _Vcvv 2681    i^i cin 3065    C_ wss 3066   dom cdm 4534   ran crn 4535    |` cres 4536   "cima 4537   Rel wrel 4539   Fun wfun 5112
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-fun 5120
This theorem is referenced by:  funimaex  5203  resfunexg  5634  resfunexgALT  6001  fnexALT  6004  suplocexprlem2b  7515  suplocexprlemlub  7525
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