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Theorem ralrnmpt 6016
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 5689 . . . 4  |-  ( A. x  e.  A  B  e.  V  ->  F  Fn  A )
3 dfsbcq 3326 . . . . 5  |-  ( w  =  ( F `  z )  ->  ( [. w  /  y ]. ps  <->  [. ( F `  z )  /  y ]. ps ) )
43ralrn 6010 . . . 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 1712 . . . . 5  |-  F/ w ps
7 nfsbc1v 3344 . . . . 5  |-  F/ y
[. w  /  y ]. ps
8 sbceq1a 3335 . . . . 5  |-  ( y  =  w  ->  ( ps 
<-> 
[. w  /  y ]. ps ) )
96, 7, 8cbvral 3077 . . . 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 4528 . . . . . . 7  |-  F/_ x
( x  e.  A  |->  B )
121, 11nfcxfr 2614 . . . . . 6  |-  F/_ x F
13 nfcv 2616 . . . . . 6  |-  F/_ x
z
1412, 13nffv 5855 . . . . 5  |-  F/_ x
( F `  z
)
15 nfv 1712 . . . . 5  |-  F/ x ps
1614, 15nfsbc 3346 . . . 4  |-  F/ x [. ( F `  z
)  /  y ]. ps
17 nfv 1712 . . . 4  |-  F/ z
[. ( F `  x )  /  y ]. ps
18 fveq2 5848 . . . . 5  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
1918sbceq1d 3329 . . . 4  |-  ( z  =  x  ->  ( [. ( F `  z
)  /  y ]. ps 
<-> 
[. ( F `  x )  /  y ]. ps ) )
2016, 17, 19cbvral 3077 . . 3  |-  ( A. z  e.  A  [. ( F `  z )  /  y ]. ps  <->  A. x  e.  A  [. ( F `  x )  /  y ]. ps )
215, 10, 203bitr3g 287 . 2  |-  ( A. x  e.  A  B  e.  V  ->  ( A. y  e.  ran  F ps  <->  A. x  e.  A  [. ( F `  x )  /  y ]. ps ) )
221fvmpt2 5939 . . . . . 6  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( F `  x
)  =  B )
2322sbceq1d 3329 . . . . 5  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( [. ( F `
 x )  / 
y ]. ps  <->  [. B  / 
y ]. ps ) )
24 ralrnmpt.2 . . . . . . 7  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
2524sbcieg 3357 . . . . . 6  |-  ( B  e.  V  ->  ( [. B  /  y ]. ps  <->  ch ) )
2625adantl 464 . . . . 5  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( [. B  / 
y ]. ps  <->  ch )
)
2723, 26bitrd 253 . . . 4  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( [. ( F `
 x )  / 
y ]. ps  <->  ch )
)
2827ralimiaa 2846 . . 3  |-  ( A. x  e.  A  B  e.  V  ->  A. x  e.  A  ( [. ( F `  x )  /  y ]. ps  <->  ch ) )
29 ralbi 2985 . . 3  |-  ( A. x  e.  A  ( [. ( F `  x
)  /  y ]. ps 
<->  ch )  ->  ( A. x  e.  A  [. ( F `  x
)  /  y ]. ps 
<-> 
A. x  e.  A  ch ) )
3028, 29syl 16 . 2  |-  ( A. x  e.  A  B  e.  V  ->  ( A. x  e.  A  [. ( F `  x )  /  y ]. ps  <->  A. x  e.  A  ch ) )
3121, 30bitrd 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   [.wsbc 3324    |-> cmpt 4497   ran crn 4989    Fn wfn 5565   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578
This theorem is referenced by:  rexrnmpt  6017  ac6num  8850  gsumwspan  16213  dfod2  16785  ordtbaslem  19856  ordtrest2lem  19871  cncmp  20059  comppfsc  20199  ptpjopn  20279  ordthmeolem  20468  tsmsfbas  20792  tsmsf1o  20813  prdsxmetlem  21037  prdsbl  21160  metdsf  21518  metdsge  21519  minveclem1  22005  minveclem3b  22009  minveclem6  22015  mbflimsup  22239  xrlimcnp  23496  minvecolem1  25988  minvecolem5  25995  minvecolem6  25996  ordtrest2NEWlem  28139  cvmsss2  28983  fin2so  30280  prdsbnd  30529  rrnequiv  30571
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