Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fipreimaOLD Unicode version

Theorem fipreimaOLD 26415
Description: Given a finite subset  A of the range of a function, there exists a finite subset of the domain whose image is  A. (Moved to fipreima 7161 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Apr-2012.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 1-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
fipreimaOLD  |-  ( ( ( F  Fn  B  /\  B  e.  M
)  /\  ( A  C_ 
ran  F  /\  A  e. 
Fin ) )  ->  E. c  e.  ( ~P B  i^i  Fin )
( F " c
)  =  A )
Distinct variable groups:    F, c    A, c    B, c    M, c

Proof of Theorem fipreimaOLD
StepHypRef Expression
1 fipreima 7161 . . 3  |-  ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  ->  E. c  e.  ( ~P B  i^i  Fin ) ( F "
c )  =  A )
213expb 1152 . 2  |-  ( ( F  Fn  B  /\  ( A  C_  ran  F  /\  A  e.  Fin ) )  ->  E. c  e.  ( ~P B  i^i  Fin ) ( F "
c )  =  A )
32adantlr 695 1  |-  ( ( ( F  Fn  B  /\  B  e.  M
)  /\  ( A  C_ 
ran  F  /\  A  e. 
Fin ) )  ->  E. c  e.  ( ~P B  i^i  Fin )
( F " c
)  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   ran crn 4690   "cima 4692    Fn wfn 5250   Fincfn 6863
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-fin 6867
  Copyright terms: Public domain W3C validator