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Theorem elmptrab 17522
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 5169 . . 3  |-  dom  F  C_  D
3 elfvdm 5554 . . 3  |-  ( Y  e.  ( F `  X )  ->  X  e.  dom  F )
42, 3sseldi 3178 . 2  |-  ( Y  e.  ( F `  X )  ->  X  e.  D )
5 simp1 955 . 2  |-  ( ( X  e.  D  /\  Y  e.  C  /\  ps )  ->  X  e.  D )
6 csbeq1 3084 . . . . . 6  |-  ( z  =  X  ->  [_ z  /  x ]_ B  = 
[_ X  /  x ]_ B )
7 dfsbcq 2993 . . . . . 6  |-  ( z  =  X  ->  ( [. z  /  x ]. [. w  /  y ]. ph  <->  [. X  /  x ]. [. w  /  y ]. ph ) )
86, 7rabeqbidv 2783 . . . . 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 2419 . . . . . . 7  |-  F/_ z { y  e.  B  |  ph }
10 nfsbc1v 3010 . . . . . . . 8  |-  F/ x [. z  /  x ]. [. w  /  y ]. ph
11 nfcsb1v 3113 . . . . . . . 8  |-  F/_ x [_ z  /  x ]_ B
1210, 11nfrab 2721 . . . . . . 7  |-  F/_ x { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph }
13 csbeq1a 3089 . . . . . . . . 9  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
14 sbceq1a 3001 . . . . . . . . 9  |-  ( x  =  z  ->  ( ph 
<-> 
[. z  /  x ]. ph ) )
1513, 14rabeqbidv 2783 . . . . . . . 8  |-  ( x  =  z  ->  { y  e.  B  |  ph }  =  { y  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. ph }
)
16 nfcv 2419 . . . . . . . . 9  |-  F/_ w [_ z  /  x ]_ B
17 nfcv 2419 . . . . . . . . 9  |-  F/_ y [_ z  /  x ]_ B
18 nfcv 2419 . . . . . . . . . 10  |-  F/_ y
z
19 nfsbc1v 3010 . . . . . . . . . 10  |-  F/ y
[. w  /  y ]. ph
2018, 19nfsbc 3012 . . . . . . . . 9  |-  F/ y
[. z  /  x ]. [. w  /  y ]. ph
21 nfv 1605 . . . . . . . . 9  |-  F/ w [. z  /  x ]. ph
22 sbceq1a 3001 . . . . . . . . . . 11  |-  ( y  =  w  ->  ( [. z  /  x ]. ph  <->  [. w  /  y ]. [. z  /  x ]. ph ) )
2322equcoms 1651 . . . . . . . . . 10  |-  ( w  =  y  ->  ( [. z  /  x ]. ph  <->  [. w  /  y ]. [. z  /  x ]. ph ) )
24 sbccom 3062 . . . . . . . . . 10  |-  ( [. z  /  x ]. [. w  /  y ]. ph  <->  [. w  / 
y ]. [. z  /  x ]. ph )
2523, 24syl6rbbr 255 . . . . . . . . 9  |-  ( w  =  y  ->  ( [. z  /  x ]. [. w  /  y ]. ph  <->  [. z  /  x ]. ph ) )
2616, 17, 20, 21, 25cbvrab 2786 . . . . . . . 8  |-  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph }  =  { y  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. ph }
2715, 26syl6eqr 2333 . . . . . . 7  |-  ( x  =  z  ->  { y  e.  B  |  ph }  =  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph }
)
289, 12, 27cbvmpt 4110 . . . . . 6  |-  ( x  e.  D  |->  { y  e.  B  |  ph } )  =  ( z  e.  D  |->  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph } )
291, 28eqtri 2303 . . . . 5  |-  F  =  ( z  e.  D  |->  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph } )
30 nfv 1605 . . . . . . . 8  |-  F/ x  z  e.  D
3111nfel1 2429 . . . . . . . 8  |-  F/ x [_ z  /  x ]_ B  e.  V
3230, 31nfim 1769 . . . . . . 7  |-  F/ x
( z  e.  D  ->  [_ z  /  x ]_ B  e.  V
)
33 eleq1 2343 . . . . . . . 8  |-  ( x  =  z  ->  (
x  e.  D  <->  z  e.  D ) )
3413eleq1d 2349 . . . . . . . 8  |-  ( x  =  z  ->  ( B  e.  V  <->  [_ z  /  x ]_ B  e.  V
) )
3533, 34imbi12d 311 . . . . . . 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 1926 . . . . . 6  |-  ( z  e.  D  ->  [_ z  /  x ]_ B  e.  V )
38 rabexg 4164 . . . . . 6  |-  ( [_ z  /  x ]_ B  e.  V  ->  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph }  e.  _V )
3937, 38syl 15 . . . . 5  |-  ( z  e.  D  ->  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph }  e.  _V )
408, 29, 39fvmpt3 5604 . . . 4  |-  ( X  e.  D  ->  ( F `  X )  =  { w  e.  [_ X  /  x ]_ B  |  [. X  /  x ]. [. w  /  y ]. ph } )
4140eleq2d 2350 . . 3  |-  ( X  e.  D  ->  ( Y  e.  ( F `  X )  <->  Y  e.  { w  e.  [_ X  /  x ]_ B  |  [. X  /  x ]. [. w  /  y ]. ph } ) )
42 dfsbcq 2993 . . . . . . 7  |-  ( w  =  Y  ->  ( [. w  /  y ]. ph  <->  [. Y  /  y ]. ph ) )
4342sbcbidv 3045 . . . . . 6  |-  ( w  =  Y  ->  ( [. X  /  x ]. [. w  /  y ]. ph  <->  [. X  /  x ]. [. Y  /  y ]. ph ) )
4443elrab 2923 . . . . 5  |-  ( Y  e.  { w  e. 
[_ X  /  x ]_ B  |  [. X  /  x ]. [. w  /  y ]. ph }  <->  ( Y  e.  [_ X  /  x ]_ B  /\  [. X  /  x ]. [. Y  /  y ]. ph ) )
4544a1i 10 . . . 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 2420 . . . . . . 7  |-  ( X  e.  D  ->  F/_ x C )
47 elmptrab.s2 . . . . . . 7  |-  ( x  =  X  ->  B  =  C )
4846, 47csbiegf 3121 . . . . . 6  |-  ( X  e.  D  ->  [_ X  /  x ]_ B  =  C )
4948eleq2d 2350 . . . . 5  |-  ( X  e.  D  ->  ( Y  e.  [_ X  /  x ]_ B  <->  Y  e.  C ) )
5049anbi1d 685 . . . 4  |-  ( X  e.  D  ->  (
( Y  e.  [_ X  /  x ]_ B  /\  [. X  /  x ]. [. Y  /  y ]. ph )  <->  ( Y  e.  C  /\  [. X  /  x ]. [. Y  /  y ]. ph )
) )
51 nfv 1605 . . . . . 6  |-  F/ x ps
52 nfv 1605 . . . . . 6  |-  F/ y ps
53 nfv 1605 . . . . . 6  |-  F/ x  Y  e.  C
54 elmptrab.s1 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ph  <->  ps )
)
5551, 52, 53, 54sbc2iegf 3057 . . . . 5  |-  ( ( X  e.  D  /\  Y  e.  C )  ->  ( [. X  /  x ]. [. Y  / 
y ]. ph  <->  ps )
)
5655pm5.32da 622 . . . 4  |-  ( X  e.  D  ->  (
( Y  e.  C  /\  [. X  /  x ]. [. Y  /  y ]. ph )  <->  ( Y  e.  C  /\  ps )
) )
5745, 50, 563bitrd 270 . . 3  |-  ( X  e.  D  ->  ( Y  e.  { w  e.  [_ X  /  x ]_ B  |  [. X  /  x ]. [. w  /  y ]. ph }  <->  ( Y  e.  C  /\  ps ) ) )
58 3anass 938 . . . 4  |-  ( ( X  e.  D  /\  Y  e.  C  /\  ps )  <->  ( X  e.  D  /\  ( Y  e.  C  /\  ps ) ) )
5958baibr 872 . . 3  |-  ( X  e.  D  ->  (
( Y  e.  C  /\  ps )  <->  ( X  e.  D  /\  Y  e.  C  /\  ps )
) )
6041, 57, 593bitrd 270 . 2  |-  ( X  e.  D  ->  ( Y  e.  ( F `  X )  <->  ( X  e.  D  /\  Y  e.  C  /\  ps )
) )
614, 5, 60pm5.21nii 342 1  |-  ( Y  e.  ( F `  X )  <->  ( X  e.  D  /\  Y  e.  C  /\  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   [.wsbc 2991   [_csb 3081    e. cmpt 4077   dom cdm 4689   ` cfv 5255
This theorem is referenced by:  elmptrab2  17523  isfbas  17524
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263
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