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Theorem ralrn 6022
Description: Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
Hypothesis
Ref Expression
rexrn.1  |-  ( x  =  ( F `  y )  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ralrn  |-  ( F  Fn  A  ->  ( A. x  e.  ran  F
ph 
<-> 
A. y  e.  A  ps ) )
Distinct variable groups:    x, y, A    x, F, y    ps, x    ph, y
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem ralrn
StepHypRef Expression
1 fvex 5874 . . 3  |-  ( F `
 y )  e. 
_V
21a1i 11 . 2  |-  ( ( F  Fn  A  /\  y  e.  A )  ->  ( F `  y
)  e.  _V )
3 fvelrnb 5913 . . 3  |-  ( F  Fn  A  ->  (
x  e.  ran  F  <->  E. y  e.  A  ( F `  y )  =  x ) )
4 eqcom 2476 . . . 4  |-  ( ( F `  y )  =  x  <->  x  =  ( F `  y ) )
54rexbii 2965 . . 3  |-  ( E. y  e.  A  ( F `  y )  =  x  <->  E. y  e.  A  x  =  ( F `  y ) )
63, 5syl6bb 261 . 2  |-  ( F  Fn  A  ->  (
x  e.  ran  F  <->  E. y  e.  A  x  =  ( F `  y ) ) )
7 rexrn.1 . . 3  |-  ( x  =  ( F `  y )  ->  ( ph 
<->  ps ) )
87adantl 466 . 2  |-  ( ( F  Fn  A  /\  x  =  ( F `  y ) )  -> 
( ph  <->  ps ) )
92, 6, 8ralxfr2d 4663 1  |-  ( F  Fn  A  ->  ( A. x  e.  ran  F
ph 
<-> 
A. y  e.  A  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113   ran crn 5000    Fn wfn 5581   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594
This theorem is referenced by:  ralrnmpt  6028  cbvfo  6178  isoselem  6223  indexfi  7824  ordtypelem9  7947  ordtypelem10  7948  wemapwe  8135  wemapweOLD  8136  numacn  8426  acndom  8428  rpnnen1lem3  11206  fsequb2  12050  limsuple  13260  limsupval2  13262  climsup  13451  ruclem11  13830  ruclem12  13831  prmreclem6  14294  imasaddfnlem  14779  imasvscafn  14788  cycsubgcl  16022  ghmrn  16075  ghmnsgima  16085  pgpssslw  16430  gexex  16652  dprdfcntz  16839  dprdfcntzOLD  16845  znf1o  18357  frlmlbs  18598  lindfrn  18623  ptcnplem  19857  kqt0lem  19972  isr0  19973  regr1lem2  19976  uzrest  20133  tmdgsum2  20330  imasf1oxmet  20613  imasf1omet  20614  bndth  21193  evth  21194  ovolficcss  21616  ovollb2lem  21634  ovolunlem1  21643  ovoliunlem1  21648  ovoliunlem2  21649  ovoliun2  21652  ovolscalem1  21659  ovolicc1  21662  voliunlem2  21696  voliunlem3  21697  ioombl1lem4  21706  uniioovol  21723  uniioombllem2  21727  uniioombllem3  21729  uniioombllem6  21732  volsup2  21749  vitalilem3  21754  mbfsup  21806  mbfinf  21807  mbflimsup  21808  itg1ge0  21828  itg1mulc  21846  itg1climres  21856  mbfi1fseqlem4  21860  itg2seq  21884  itg2monolem1  21892  itg2mono  21895  itg2i1fseq2  21898  itg2gt0  21902  itg2cnlem1  21903  itg2cn  21905  limciun  22033  plycpn  22419  hmopidmchi  26746  hmopidmpji  26747  rge0scvg  27567  mblfinlem2  29629  ismtyhmeolem  29903  nacsfix  30248  fnwe2lem2  30601  climinf  31148
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