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Theorem elmptrab 20777
Description: Membership in a one-parameter class of sets. (Contributed by Stefan O'Rear, 28-Jul-2015.)
Hypotheses
Ref Expression
elmptrab.f  |-  F  =  ( x  e.  D  |->  { y  e.  B  |  ph } )
elmptrab.s1  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ph  <->  ps )
)
elmptrab.s2  |-  ( x  =  X  ->  B  =  C )
elmptrab.ex  |-  ( x  e.  D  ->  B  e.  V )
Assertion
Ref Expression
elmptrab  |-  ( Y  e.  ( F `  X )  <->  ( X  e.  D  /\  Y  e.  C  /\  ps )
)
Distinct variable groups:    x, y, X    y, B    x, C, y    x, D    x, V, y    x, Y, y    ps, x, y
Allowed substitution hints:    ph( x, y)    B( x)    D( y)    F( x, y)

Proof of Theorem elmptrab
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elmptrab.f . . 3  |-  F  =  ( x  e.  D  |->  { y  e.  B  |  ph } )
21mptrcl 5908 . 2  |-  ( Y  e.  ( F `  X )  ->  X  e.  D )
3 simp1 1005 . 2  |-  ( ( X  e.  D  /\  Y  e.  C  /\  ps )  ->  X  e.  D )
4 csbeq1 3334 . . . . . 6  |-  ( z  =  X  ->  [_ z  /  x ]_ B  = 
[_ X  /  x ]_ B )
5 dfsbcq 3237 . . . . . 6  |-  ( z  =  X  ->  ( [. z  /  x ]. [. w  /  y ]. ph  <->  [. X  /  x ]. [. w  /  y ]. ph ) )
64, 5rabeqbidv 3011 . . . . 5  |-  ( z  =  X  ->  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph }  =  { w  e.  [_ X  /  x ]_ B  |  [. X  /  x ]. [. w  /  y ]. ph } )
7 nfcv 2563 . . . . . . 7  |-  F/_ z { y  e.  B  |  ph }
8 nfsbc1v 3255 . . . . . . . 8  |-  F/ x [. z  /  x ]. [. w  /  y ]. ph
9 nfcsb1v 3347 . . . . . . . 8  |-  F/_ x [_ z  /  x ]_ B
108, 9nfrab 2943 . . . . . . 7  |-  F/_ x { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph }
11 csbeq1a 3340 . . . . . . . . 9  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
12 sbceq1a 3246 . . . . . . . . 9  |-  ( x  =  z  ->  ( ph 
<-> 
[. z  /  x ]. ph ) )
1311, 12rabeqbidv 3011 . . . . . . . 8  |-  ( x  =  z  ->  { y  e.  B  |  ph }  =  { y  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. ph }
)
14 nfcv 2563 . . . . . . . . 9  |-  F/_ w [_ z  /  x ]_ B
15 nfcv 2563 . . . . . . . . 9  |-  F/_ y [_ z  /  x ]_ B
16 nfcv 2563 . . . . . . . . . 10  |-  F/_ y
z
17 nfsbc1v 3255 . . . . . . . . . 10  |-  F/ y
[. w  /  y ]. ph
1816, 17nfsbc 3257 . . . . . . . . 9  |-  F/ y
[. z  /  x ]. [. w  /  y ]. ph
19 nfv 1755 . . . . . . . . 9  |-  F/ w [. z  /  x ]. ph
20 sbceq1a 3246 . . . . . . . . . . 11  |-  ( y  =  w  ->  ( [. z  /  x ]. ph  <->  [. w  /  y ]. [. z  /  x ]. ph ) )
2120equcoms 1849 . . . . . . . . . 10  |-  ( w  =  y  ->  ( [. z  /  x ]. ph  <->  [. w  /  y ]. [. z  /  x ]. ph ) )
22 sbccom 3307 . . . . . . . . . 10  |-  ( [. z  /  x ]. [. w  /  y ]. ph  <->  [. w  / 
y ]. [. z  /  x ]. ph )
2321, 22syl6rbbr 267 . . . . . . . . 9  |-  ( w  =  y  ->  ( [. z  /  x ]. [. w  /  y ]. ph  <->  [. z  /  x ]. ph ) )
2414, 15, 18, 19, 23cbvrab 3014 . . . . . . . 8  |-  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph }  =  { y  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. ph }
2513, 24syl6eqr 2474 . . . . . . 7  |-  ( x  =  z  ->  { y  e.  B  |  ph }  =  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph }
)
267, 10, 25cbvmpt 4451 . . . . . 6  |-  ( x  e.  D  |->  { y  e.  B  |  ph } )  =  ( z  e.  D  |->  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph } )
271, 26eqtri 2444 . . . . 5  |-  F  =  ( z  e.  D  |->  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph } )
28 nfv 1755 . . . . . . . 8  |-  F/ x  z  e.  D
299nfel1 2577 . . . . . . . 8  |-  F/ x [_ z  /  x ]_ B  e.  V
3028, 29nfim 1980 . . . . . . 7  |-  F/ x
( z  e.  D  ->  [_ z  /  x ]_ B  e.  V
)
31 eleq1 2488 . . . . . . . 8  |-  ( x  =  z  ->  (
x  e.  D  <->  z  e.  D ) )
3211eleq1d 2484 . . . . . . . 8  |-  ( x  =  z  ->  ( B  e.  V  <->  [_ z  /  x ]_ B  e.  V
) )
3331, 32imbi12d 321 . . . . . . 7  |-  ( x  =  z  ->  (
( x  e.  D  ->  B  e.  V )  <-> 
( z  e.  D  ->  [_ z  /  x ]_ B  e.  V
) ) )
34 elmptrab.ex . . . . . . 7  |-  ( x  e.  D  ->  B  e.  V )
3530, 33, 34chvar 2072 . . . . . 6  |-  ( z  e.  D  ->  [_ z  /  x ]_ B  e.  V )
36 rabexg 4510 . . . . . 6  |-  ( [_ z  /  x ]_ B  e.  V  ->  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph }  e.  _V )
3735, 36syl 17 . . . . 5  |-  ( z  e.  D  ->  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph }  e.  _V )
386, 27, 37fvmpt3 5905 . . . 4  |-  ( X  e.  D  ->  ( F `  X )  =  { w  e.  [_ X  /  x ]_ B  |  [. X  /  x ]. [. w  /  y ]. ph } )
3938eleq2d 2485 . . 3  |-  ( X  e.  D  ->  ( Y  e.  ( F `  X )  <->  Y  e.  { w  e.  [_ X  /  x ]_ B  |  [. X  /  x ]. [. w  /  y ]. ph } ) )
40 dfsbcq 3237 . . . . . . 7  |-  ( w  =  Y  ->  ( [. w  /  y ]. ph  <->  [. Y  /  y ]. ph ) )
4140sbcbidv 3290 . . . . . 6  |-  ( w  =  Y  ->  ( [. X  /  x ]. [. w  /  y ]. ph  <->  [. X  /  x ]. [. Y  /  y ]. ph ) )
4241elrab 3164 . . . . 5  |-  ( Y  e.  { w  e. 
[_ X  /  x ]_ B  |  [. X  /  x ]. [. w  /  y ]. ph }  <->  ( Y  e.  [_ X  /  x ]_ B  /\  [. X  /  x ]. [. Y  /  y ]. ph ) )
4342a1i 11 . . . 4  |-  ( X  e.  D  ->  ( Y  e.  { w  e.  [_ X  /  x ]_ B  |  [. X  /  x ]. [. w  /  y ]. ph }  <->  ( Y  e.  [_ X  /  x ]_ B  /\  [. X  /  x ]. [. Y  /  y ]. ph ) ) )
44 nfcvd 2564 . . . . . . 7  |-  ( X  e.  D  ->  F/_ x C )
45 elmptrab.s2 . . . . . . 7  |-  ( x  =  X  ->  B  =  C )
4644, 45csbiegf 3355 . . . . . 6  |-  ( X  e.  D  ->  [_ X  /  x ]_ B  =  C )
4746eleq2d 2485 . . . . 5  |-  ( X  e.  D  ->  ( Y  e.  [_ X  /  x ]_ B  <->  Y  e.  C ) )
4847anbi1d 709 . . . 4  |-  ( X  e.  D  ->  (
( Y  e.  [_ X  /  x ]_ B  /\  [. X  /  x ]. [. Y  /  y ]. ph )  <->  ( Y  e.  C  /\  [. X  /  x ]. [. Y  /  y ]. ph )
) )
49 nfv 1755 . . . . . 6  |-  F/ x ps
50 nfv 1755 . . . . . 6  |-  F/ y ps
51 nfv 1755 . . . . . 6  |-  F/ x  Y  e.  C
52 elmptrab.s1 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ph  <->  ps )
)
5349, 50, 51, 52sbc2iegf 3302 . . . . 5  |-  ( ( X  e.  D  /\  Y  e.  C )  ->  ( [. X  /  x ]. [. Y  / 
y ]. ph  <->  ps )
)
5453pm5.32da 645 . . . 4  |-  ( X  e.  D  ->  (
( Y  e.  C  /\  [. X  /  x ]. [. Y  /  y ]. ph )  <->  ( Y  e.  C  /\  ps )
) )
5543, 48, 543bitrd 282 . . 3  |-  ( X  e.  D  ->  ( Y  e.  { w  e.  [_ X  /  x ]_ B  |  [. X  /  x ]. [. w  /  y ]. ph }  <->  ( Y  e.  C  /\  ps ) ) )
56 3anass 986 . . . 4  |-  ( ( X  e.  D  /\  Y  e.  C  /\  ps )  <->  ( X  e.  D  /\  ( Y  e.  C  /\  ps ) ) )
5756baibr 912 . . 3  |-  ( X  e.  D  ->  (
( Y  e.  C  /\  ps )  <->  ( X  e.  D  /\  Y  e.  C  /\  ps )
) )
5839, 55, 573bitrd 282 . 2  |-  ( X  e.  D  ->  ( Y  e.  ( F `  X )  <->  ( X  e.  D  /\  Y  e.  C  /\  ps )
) )
592, 3, 58pm5.21nii 354 1  |-  ( Y  e.  ( F `  X )  <->  ( X  e.  D  /\  Y  e.  C  /\  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   {crab 2712   _Vcvv 3016   [.wsbc 3235   [_csb 3331    |-> cmpt 4418   ` cfv 5537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-sep 4482  ax-nul 4491  ax-pow 4538  ax-pr 4596
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-ral 2713  df-rex 2714  df-rab 2717  df-v 3018  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3698  df-if 3848  df-sn 3935  df-pr 3937  df-op 3941  df-uni 4156  df-br 4360  df-opab 4419  df-mpt 4420  df-id 4704  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-res 4801  df-ima 4802  df-iota 5501  df-fun 5539  df-fv 5545
This theorem is referenced by:  elmptrab2  20778  isfbas  20779
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