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Theorem funimass5 6005
Description: A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.)
Assertion
Ref Expression
funimass5  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( A  C_  ( `' F " B )  <->  A. x  e.  A  ( F `  x )  e.  B ) )
Distinct variable groups:    x, F    x, A    x, B

Proof of Theorem funimass5
StepHypRef Expression
1 funimass3 6004 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A 
C_  ( `' F " B ) ) )
2 funimass4 5924 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A. x  e.  A  ( F `  x )  e.  B ) )
31, 2bitr3d 255 1  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( A  C_  ( `' F " B )  <->  A. x  e.  A  ( F `  x )  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1819   A.wral 2807    C_ wss 3471   `'ccnv 5007   dom cdm 5008   "cima 5011   Fun wfun 5588   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602
This theorem is referenced by:  clssubg  20733  mclspps  29141
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