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Theorem funimaexg 5084
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 107 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  Fun  A )
2 funrel 5019 . . 3  |-  ( Fun 
A  ->  Rel  A )
3 resres 4713 . . . . . . 7  |-  ( ( A  |`  dom  A )  |`  B )  =  ( A  |`  ( dom  A  i^i  B ) )
4 incom 3190 . . . . . . . 8  |-  ( B  i^i  dom  A )  =  ( dom  A  i^i  B )
54reseq2i 4698 . . . . . . 7  |-  ( A  |`  ( B  i^i  dom  A ) )  =  ( A  |`  ( dom  A  i^i  B ) )
63, 5eqtr4i 2111 . . . . . 6  |-  ( ( A  |`  dom  A )  |`  B )  =  ( A  |`  ( B  i^i  dom  A ) )
7 resdm 4738 . . . . . . 7  |-  ( Rel 
A  ->  ( A  |` 
dom  A )  =  A )
87reseq1d 4700 . . . . . 6  |-  ( Rel 
A  ->  ( ( A  |`  dom  A )  |`  B )  =  ( A  |`  B )
)
96, 8syl5eqr 2134 . . . . 5  |-  ( Rel 
A  ->  ( A  |`  ( B  i^i  dom  A ) )  =  ( A  |`  B )
)
109rneqd 4652 . . . 4  |-  ( Rel 
A  ->  ran  ( A  |`  ( B  i^i  dom  A ) )  =  ran  ( A  |`  B ) )
11 df-ima 4441 . . . 4  |-  ( A
" ( B  i^i  dom 
A ) )  =  ran  ( A  |`  ( B  i^i  dom  A
) )
12 df-ima 4441 . . . 4  |-  ( A
" B )  =  ran  ( A  |`  B )
1310, 11, 123eqtr4g 2145 . . 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 3967 . . 3  |-  ( B  e.  C  ->  ( B  i^i  dom  A )  e.  _V )
16 inss2 3219 . . . 4  |-  ( B  i^i  dom  A )  C_ 
dom  A
17 funimaexglem 5083 . . . 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 1262 . . 3  |-  ( ( Fun  A  /\  ( B  i^i  dom  A )  e.  _V )  ->  ( A " ( B  i^i  dom 
A ) )  e. 
_V )
1915, 18sylan2 280 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A " ( B  i^i  dom 
A ) )  e. 
_V )
2014, 19eqeltrrd 2165 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A " B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   _Vcvv 2619    i^i cin 2996    C_ wss 2997   dom cdm 4428   ran crn 4429    |` cres 4430   "cima 4431   Rel wrel 4433   Fun wfun 4996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-fun 5004
This theorem is referenced by:  funimaex  5085  resfunexg  5500  resfunexgALT  5863  fnexALT  5866
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