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Theorem ralrnmpt 6029
Description: A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)
Hypotheses
Ref Expression
ralrnmpt.1  |-  F  =  ( x  e.  A  |->  B )
ralrnmpt.2  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
ralrnmpt  |-  ( A. x  e.  A  B  e.  V  ->  ( A. y  e.  ran  F ps  <->  A. x  e.  A  ch ) )
Distinct variable groups:    x, A    y, B    ch, y    y, F    ps, x
Allowed substitution hints:    ps( y)    ch( x)    A( y)    B( x)    F( x)    V( x, y)

Proof of Theorem ralrnmpt
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ralrnmpt.1 . . . . 5  |-  F  =  ( x  e.  A  |->  B )
21fnmpt 5706 . . . 4  |-  ( A. x  e.  A  B  e.  V  ->  F  Fn  A )
3 dfsbcq 3333 . . . . 5  |-  ( w  =  ( F `  z )  ->  ( [. w  /  y ]. ps  <->  [. ( F `  z )  /  y ]. ps ) )
43ralrn 6023 . . . 4  |-  ( F  Fn  A  ->  ( A. w  e.  ran  F
[. w  /  y ]. ps  <->  A. z  e.  A  [. ( F `  z
)  /  y ]. ps ) )
52, 4syl 16 . . 3  |-  ( A. x  e.  A  B  e.  V  ->  ( A. w  e.  ran  F [. w  /  y ]. ps  <->  A. z  e.  A  [. ( F `  z )  /  y ]. ps ) )
6 nfv 1683 . . . . 5  |-  F/ w ps
7 nfsbc1v 3351 . . . . 5  |-  F/ y
[. w  /  y ]. ps
8 sbceq1a 3342 . . . . 5  |-  ( y  =  w  ->  ( ps 
<-> 
[. w  /  y ]. ps ) )
96, 7, 8cbvral 3084 . . . 4  |-  ( A. y  e.  ran  F ps  <->  A. w  e.  ran  F [. w  /  y ]. ps )
109bicomi 202 . . 3  |-  ( A. w  e.  ran  F [. w  /  y ]. ps  <->  A. y  e.  ran  F ps )
11 nfmpt1 4536 . . . . . . 7  |-  F/_ x
( x  e.  A  |->  B )
121, 11nfcxfr 2627 . . . . . 6  |-  F/_ x F
13 nfcv 2629 . . . . . 6  |-  F/_ x
z
1412, 13nffv 5872 . . . . 5  |-  F/_ x
( F `  z
)
15 nfv 1683 . . . . 5  |-  F/ x ps
1614, 15nfsbc 3353 . . . 4  |-  F/ x [. ( F `  z
)  /  y ]. ps
17 nfv 1683 . . . 4  |-  F/ z
[. ( F `  x )  /  y ]. ps
18 fveq2 5865 . . . . 5  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
19 dfsbcq 3333 . . . . 5  |-  ( ( F `  z )  =  ( F `  x )  ->  ( [. ( F `  z
)  /  y ]. ps 
<-> 
[. ( F `  x )  /  y ]. ps ) )
2018, 19syl 16 . . . 4  |-  ( z  =  x  ->  ( [. ( F `  z
)  /  y ]. ps 
<-> 
[. ( F `  x )  /  y ]. ps ) )
2116, 17, 20cbvral 3084 . . 3  |-  ( A. z  e.  A  [. ( F `  z )  /  y ]. ps  <->  A. x  e.  A  [. ( F `  x )  /  y ]. ps )
225, 10, 213bitr3g 287 . 2  |-  ( A. x  e.  A  B  e.  V  ->  ( A. y  e.  ran  F ps  <->  A. x  e.  A  [. ( F `  x )  /  y ]. ps ) )
231fvmpt2 5956 . . . . . 6  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( F `  x
)  =  B )
24 dfsbcq 3333 . . . . . 6  |-  ( ( F `  x )  =  B  ->  ( [. ( F `  x
)  /  y ]. ps 
<-> 
[. B  /  y ]. ps ) )
2523, 24syl 16 . . . . 5  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( [. ( F `
 x )  / 
y ]. ps  <->  [. B  / 
y ]. ps ) )
26 ralrnmpt.2 . . . . . . 7  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
2726sbcieg 3364 . . . . . 6  |-  ( B  e.  V  ->  ( [. B  /  y ]. ps  <->  ch ) )
2827adantl 466 . . . . 5  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( [. B  / 
y ]. ps  <->  ch )
)
2925, 28bitrd 253 . . . 4  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( [. ( F `
 x )  / 
y ]. ps  <->  ch )
)
3029ralimiaa 2856 . . 3  |-  ( A. x  e.  A  B  e.  V  ->  A. x  e.  A  ( [. ( F `  x )  /  y ]. ps  <->  ch ) )
31 ralbi 2993 . . 3  |-  ( A. x  e.  A  ( [. ( F `  x
)  /  y ]. ps 
<->  ch )  ->  ( A. x  e.  A  [. ( F `  x
)  /  y ]. ps 
<-> 
A. x  e.  A  ch ) )
3230, 31syl 16 . 2  |-  ( A. x  e.  A  B  e.  V  ->  ( A. x  e.  A  [. ( F `  x )  /  y ]. ps  <->  A. x  e.  A  ch ) )
3322, 32bitrd 253 1  |-  ( A. x  e.  A  B  e.  V  ->  ( A. y  e.  ran  F ps  <->  A. x  e.  A  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   [.wsbc 3331    |-> cmpt 4505   ran crn 5000    Fn wfn 5582   ` cfv 5587
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-8 1769  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-pow 4625  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-csb 3436  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-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-fv 5595
This theorem is referenced by:  rexrnmpt  6030  ac6num  8858  gsumwspan  15843  dfod2  16389  ordtbaslem  19471  ordtrest2lem  19486  cncmp  19674  ptpjopn  19864  ordthmeolem  20053  tsmsfbas  20377  tsmsf1o  20398  prdsxmetlem  20622  prdsbl  20745  metdsf  21103  metdsge  21104  minveclem1  21590  minveclem3b  21594  minveclem6  21600  mbflimsup  21824  xrlimcnp  23042  minvecolem1  25482  minvecolem5  25489  minvecolem6  25490  ordtrest2NEWlem  27556  cvmsss2  28375  fin2so  29633  comppfsc  29795  prdsbnd  29908  rrnequiv  29950
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