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Theorem funimaexg 5034
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 4969 . . 3  |-  ( Fun 
A  ->  Rel  A )
3 resres 4672 . . . . . . 7  |-  ( ( A  |`  dom  A )  |`  B )  =  ( A  |`  ( dom  A  i^i  B ) )
4 incom 3174 . . . . . . . 8  |-  ( B  i^i  dom  A )  =  ( dom  A  i^i  B )
54reseq2i 4657 . . . . . . 7  |-  ( A  |`  ( B  i^i  dom  A ) )  =  ( A  |`  ( dom  A  i^i  B ) )
63, 5eqtr4i 2106 . . . . . 6  |-  ( ( A  |`  dom  A )  |`  B )  =  ( A  |`  ( B  i^i  dom  A ) )
7 resdm 4697 . . . . . . 7  |-  ( Rel 
A  ->  ( A  |` 
dom  A )  =  A )
87reseq1d 4659 . . . . . 6  |-  ( Rel 
A  ->  ( ( A  |`  dom  A )  |`  B )  =  ( A  |`  B )
)
96, 8syl5eqr 2129 . . . . 5  |-  ( Rel 
A  ->  ( A  |`  ( B  i^i  dom  A ) )  =  ( A  |`  B )
)
109rneqd 4611 . . . 4  |-  ( Rel 
A  ->  ran  ( A  |`  ( B  i^i  dom  A ) )  =  ran  ( A  |`  B ) )
11 df-ima 4404 . . . 4  |-  ( A
" ( B  i^i  dom 
A ) )  =  ran  ( A  |`  ( B  i^i  dom  A
) )
12 df-ima 4404 . . . 4  |-  ( A
" B )  =  ran  ( A  |`  B )
1310, 11, 123eqtr4g 2140 . . 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 3934 . . 3  |-  ( B  e.  C  ->  ( B  i^i  dom  A )  e.  _V )
16 inss2 3203 . . . 4  |-  ( B  i^i  dom  A )  C_ 
dom  A
17 funimaexglem 5033 . . . 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 1258 . . 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 2160 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A " B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   _Vcvv 2610    i^i cin 2981    C_ wss 2982   dom cdm 4391   ran crn 4392    |` cres 4393   "cima 4394   Rel wrel 4396   Fun wfun 4946
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-fun 4954
This theorem is referenced by:  funimaex  5035  resfunexg  5434  resfunexgALT  5788  fnexALT  5791
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