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Theorem elmptrab 17812
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 . . . 4  |-  F  =  ( x  e.  D  |->  { y  e.  B  |  ph } )
21dmmptss 5325 . . 3  |-  dom  F  C_  D
3 elfvdm 5716 . . 3  |-  ( Y  e.  ( F `  X )  ->  X  e.  dom  F )
42, 3sseldi 3306 . 2  |-  ( Y  e.  ( F `  X )  ->  X  e.  D )
5 simp1 957 . 2  |-  ( ( X  e.  D  /\  Y  e.  C  /\  ps )  ->  X  e.  D )
6 csbeq1 3214 . . . . . 6  |-  ( z  =  X  ->  [_ z  /  x ]_ B  = 
[_ X  /  x ]_ B )
7 dfsbcq 3123 . . . . . 6  |-  ( z  =  X  ->  ( [. z  /  x ]. [. w  /  y ]. ph  <->  [. X  /  x ]. [. w  /  y ]. ph ) )
86, 7rabeqbidv 2911 . . . . 5  |-  ( z  =  X  ->  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph }  =  { w  e.  [_ X  /  x ]_ B  |  [. X  /  x ]. [. w  /  y ]. ph } )
9 nfcv 2540 . . . . . . 7  |-  F/_ z { y  e.  B  |  ph }
10 nfsbc1v 3140 . . . . . . . 8  |-  F/ x [. z  /  x ]. [. w  /  y ]. ph
11 nfcsb1v 3243 . . . . . . . 8  |-  F/_ x [_ z  /  x ]_ B
1210, 11nfrab 2849 . . . . . . 7  |-  F/_ x { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph }
13 csbeq1a 3219 . . . . . . . . 9  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
14 sbceq1a 3131 . . . . . . . . 9  |-  ( x  =  z  ->  ( ph 
<-> 
[. z  /  x ]. ph ) )
1513, 14rabeqbidv 2911 . . . . . . . 8  |-  ( x  =  z  ->  { y  e.  B  |  ph }  =  { y  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. ph }
)
16 nfcv 2540 . . . . . . . . 9  |-  F/_ w [_ z  /  x ]_ B
17 nfcv 2540 . . . . . . . . 9  |-  F/_ y [_ z  /  x ]_ B
18 nfcv 2540 . . . . . . . . . 10  |-  F/_ y
z
19 nfsbc1v 3140 . . . . . . . . . 10  |-  F/ y
[. w  /  y ]. ph
2018, 19nfsbc 3142 . . . . . . . . 9  |-  F/ y
[. z  /  x ]. [. w  /  y ]. ph
21 nfv 1626 . . . . . . . . 9  |-  F/ w [. z  /  x ]. ph
22 sbceq1a 3131 . . . . . . . . . . 11  |-  ( y  =  w  ->  ( [. z  /  x ]. ph  <->  [. w  /  y ]. [. z  /  x ]. ph ) )
2322equcoms 1689 . . . . . . . . . 10  |-  ( w  =  y  ->  ( [. z  /  x ]. ph  <->  [. w  /  y ]. [. z  /  x ]. ph ) )
24 sbccom 3192 . . . . . . . . . 10  |-  ( [. z  /  x ]. [. w  /  y ]. ph  <->  [. w  / 
y ]. [. z  /  x ]. ph )
2523, 24syl6rbbr 256 . . . . . . . . 9  |-  ( w  =  y  ->  ( [. z  /  x ]. [. w  /  y ]. ph  <->  [. z  /  x ]. ph ) )
2616, 17, 20, 21, 25cbvrab 2914 . . . . . . . 8  |-  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph }  =  { y  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. ph }
2715, 26syl6eqr 2454 . . . . . . 7  |-  ( x  =  z  ->  { y  e.  B  |  ph }  =  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph }
)
289, 12, 27cbvmpt 4259 . . . . . 6  |-  ( x  e.  D  |->  { y  e.  B  |  ph } )  =  ( z  e.  D  |->  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph } )
291, 28eqtri 2424 . . . . 5  |-  F  =  ( z  e.  D  |->  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph } )
30 nfv 1626 . . . . . . . 8  |-  F/ x  z  e.  D
3111nfel1 2550 . . . . . . . 8  |-  F/ x [_ z  /  x ]_ B  e.  V
3230, 31nfim 1828 . . . . . . 7  |-  F/ x
( z  e.  D  ->  [_ z  /  x ]_ B  e.  V
)
33 eleq1 2464 . . . . . . . 8  |-  ( x  =  z  ->  (
x  e.  D  <->  z  e.  D ) )
3413eleq1d 2470 . . . . . . . 8  |-  ( x  =  z  ->  ( B  e.  V  <->  [_ z  /  x ]_ B  e.  V
) )
3533, 34imbi12d 312 . . . . . . 7  |-  ( x  =  z  ->  (
( x  e.  D  ->  B  e.  V )  <-> 
( z  e.  D  ->  [_ z  /  x ]_ B  e.  V
) ) )
36 elmptrab.ex . . . . . . 7  |-  ( x  e.  D  ->  B  e.  V )
3732, 35, 36chvar 2039 . . . . . 6  |-  ( z  e.  D  ->  [_ z  /  x ]_ B  e.  V )
38 rabexg 4313 . . . . . 6  |-  ( [_ z  /  x ]_ B  e.  V  ->  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph }  e.  _V )
3937, 38syl 16 . . . . 5  |-  ( z  e.  D  ->  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph }  e.  _V )
408, 29, 39fvmpt3 5767 . . . 4  |-  ( X  e.  D  ->  ( F `  X )  =  { w  e.  [_ X  /  x ]_ B  |  [. X  /  x ]. [. w  /  y ]. ph } )
4140eleq2d 2471 . . 3  |-  ( X  e.  D  ->  ( Y  e.  ( F `  X )  <->  Y  e.  { w  e.  [_ X  /  x ]_ B  |  [. X  /  x ]. [. w  /  y ]. ph } ) )
42 dfsbcq 3123 . . . . . . 7  |-  ( w  =  Y  ->  ( [. w  /  y ]. ph  <->  [. Y  /  y ]. ph ) )
4342sbcbidv 3175 . . . . . 6  |-  ( w  =  Y  ->  ( [. X  /  x ]. [. w  /  y ]. ph  <->  [. X  /  x ]. [. Y  /  y ]. ph ) )
4443elrab 3052 . . . . 5  |-  ( Y  e.  { w  e. 
[_ X  /  x ]_ B  |  [. X  /  x ]. [. w  /  y ]. ph }  <->  ( Y  e.  [_ X  /  x ]_ B  /\  [. X  /  x ]. [. Y  /  y ]. ph ) )
4544a1i 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 ) ) )
46 nfcvd 2541 . . . . . . 7  |-  ( X  e.  D  ->  F/_ x C )
47 elmptrab.s2 . . . . . . 7  |-  ( x  =  X  ->  B  =  C )
4846, 47csbiegf 3251 . . . . . 6  |-  ( X  e.  D  ->  [_ X  /  x ]_ B  =  C )
4948eleq2d 2471 . . . . 5  |-  ( X  e.  D  ->  ( Y  e.  [_ X  /  x ]_ B  <->  Y  e.  C ) )
5049anbi1d 686 . . . 4  |-  ( X  e.  D  ->  (
( Y  e.  [_ X  /  x ]_ B  /\  [. X  /  x ]. [. Y  /  y ]. ph )  <->  ( Y  e.  C  /\  [. X  /  x ]. [. Y  /  y ]. ph )
) )
51 nfv 1626 . . . . . 6  |-  F/ x ps
52 nfv 1626 . . . . . 6  |-  F/ y ps
53 nfv 1626 . . . . . 6  |-  F/ x  Y  e.  C
54 elmptrab.s1 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ph  <->  ps )
)
5551, 52, 53, 54sbc2iegf 3187 . . . . 5  |-  ( ( X  e.  D  /\  Y  e.  C )  ->  ( [. X  /  x ]. [. Y  / 
y ]. ph  <->  ps )
)
5655pm5.32da 623 . . . 4  |-  ( X  e.  D  ->  (
( Y  e.  C  /\  [. X  /  x ]. [. Y  /  y ]. ph )  <->  ( Y  e.  C  /\  ps )
) )
5745, 50, 563bitrd 271 . . 3  |-  ( X  e.  D  ->  ( Y  e.  { w  e.  [_ X  /  x ]_ B  |  [. X  /  x ]. [. w  /  y ]. ph }  <->  ( Y  e.  C  /\  ps ) ) )
58 3anass 940 . . . 4  |-  ( ( X  e.  D  /\  Y  e.  C  /\  ps )  <->  ( X  e.  D  /\  ( Y  e.  C  /\  ps ) ) )
5958baibr 873 . . 3  |-  ( X  e.  D  ->  (
( Y  e.  C  /\  ps )  <->  ( X  e.  D  /\  Y  e.  C  /\  ps )
) )
6041, 57, 593bitrd 271 . 2  |-  ( X  e.  D  ->  ( Y  e.  ( F `  X )  <->  ( X  e.  D  /\  Y  e.  C  /\  ps )
) )
614, 5, 60pm5.21nii 343 1  |-  ( Y  e.  ( F `  X )  <->  ( X  e.  D  /\  Y  e.  C  /\  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   {crab 2670   _Vcvv 2916   [.wsbc 3121   [_csb 3211    e. cmpt 4226   dom cdm 4837   ` cfv 5413
This theorem is referenced by:  elmptrab2  17813  isfbas  17814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fv 5421
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