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Theorem elmptrab 20091
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 5503 . . 3  |-  dom  F  C_  D
3 elfvdm 5892 . . 3  |-  ( Y  e.  ( F `  X )  ->  X  e.  dom  F )
42, 3sseldi 3502 . 2  |-  ( Y  e.  ( F `  X )  ->  X  e.  D )
5 simp1 996 . 2  |-  ( ( X  e.  D  /\  Y  e.  C  /\  ps )  ->  X  e.  D )
6 csbeq1 3438 . . . . . 6  |-  ( z  =  X  ->  [_ z  /  x ]_ B  = 
[_ X  /  x ]_ B )
7 dfsbcq 3333 . . . . . 6  |-  ( z  =  X  ->  ( [. z  /  x ]. [. w  /  y ]. ph  <->  [. X  /  x ]. [. w  /  y ]. ph ) )
86, 7rabeqbidv 3108 . . . . 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 2629 . . . . . . 7  |-  F/_ z { y  e.  B  |  ph }
10 nfsbc1v 3351 . . . . . . . 8  |-  F/ x [. z  /  x ]. [. w  /  y ]. ph
11 nfcsb1v 3451 . . . . . . . 8  |-  F/_ x [_ z  /  x ]_ B
1210, 11nfrab 3043 . . . . . . 7  |-  F/_ x { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph }
13 csbeq1a 3444 . . . . . . . . 9  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
14 sbceq1a 3342 . . . . . . . . 9  |-  ( x  =  z  ->  ( ph 
<-> 
[. z  /  x ]. ph ) )
1513, 14rabeqbidv 3108 . . . . . . . 8  |-  ( x  =  z  ->  { y  e.  B  |  ph }  =  { y  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. ph }
)
16 nfcv 2629 . . . . . . . . 9  |-  F/_ w [_ z  /  x ]_ B
17 nfcv 2629 . . . . . . . . 9  |-  F/_ y [_ z  /  x ]_ B
18 nfcv 2629 . . . . . . . . . 10  |-  F/_ y
z
19 nfsbc1v 3351 . . . . . . . . . 10  |-  F/ y
[. w  /  y ]. ph
2018, 19nfsbc 3353 . . . . . . . . 9  |-  F/ y
[. z  /  x ]. [. w  /  y ]. ph
21 nfv 1683 . . . . . . . . 9  |-  F/ w [. z  /  x ]. ph
22 sbceq1a 3342 . . . . . . . . . . 11  |-  ( y  =  w  ->  ( [. z  /  x ]. ph  <->  [. w  /  y ]. [. z  /  x ]. ph ) )
2322equcoms 1744 . . . . . . . . . 10  |-  ( w  =  y  ->  ( [. z  /  x ]. ph  <->  [. w  /  y ]. [. z  /  x ]. ph ) )
24 sbccom 3411 . . . . . . . . . 10  |-  ( [. z  /  x ]. [. w  /  y ]. ph  <->  [. w  / 
y ]. [. z  /  x ]. ph )
2523, 24syl6rbbr 264 . . . . . . . . 9  |-  ( w  =  y  ->  ( [. z  /  x ]. [. w  /  y ]. ph  <->  [. z  /  x ]. ph ) )
2616, 17, 20, 21, 25cbvrab 3111 . . . . . . . 8  |-  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph }  =  { y  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. ph }
2715, 26syl6eqr 2526 . . . . . . 7  |-  ( x  =  z  ->  { y  e.  B  |  ph }  =  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph }
)
289, 12, 27cbvmpt 4537 . . . . . 6  |-  ( x  e.  D  |->  { y  e.  B  |  ph } )  =  ( z  e.  D  |->  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph } )
291, 28eqtri 2496 . . . . 5  |-  F  =  ( z  e.  D  |->  { w  e.  [_ z  /  x ]_ B  |  [. z  /  x ]. [. w  /  y ]. ph } )
30 nfv 1683 . . . . . . . 8  |-  F/ x  z  e.  D
3111nfel1 2645 . . . . . . . 8  |-  F/ x [_ z  /  x ]_ B  e.  V
3230, 31nfim 1867 . . . . . . 7  |-  F/ x
( z  e.  D  ->  [_ z  /  x ]_ B  e.  V
)
33 eleq1 2539 . . . . . . . 8  |-  ( x  =  z  ->  (
x  e.  D  <->  z  e.  D ) )
3413eleq1d 2536 . . . . . . . 8  |-  ( x  =  z  ->  ( B  e.  V  <->  [_ z  /  x ]_ B  e.  V
) )
3533, 34imbi12d 320 . . . . . . 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 1982 . . . . . 6  |-  ( z  e.  D  ->  [_ z  /  x ]_ B  e.  V )
38 rabexg 4597 . . . . . 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 5953 . . . 4  |-  ( X  e.  D  ->  ( F `  X )  =  { w  e.  [_ X  /  x ]_ B  |  [. X  /  x ]. [. w  /  y ]. ph } )
4140eleq2d 2537 . . 3  |-  ( X  e.  D  ->  ( Y  e.  ( F `  X )  <->  Y  e.  { w  e.  [_ X  /  x ]_ B  |  [. X  /  x ]. [. w  /  y ]. ph } ) )
42 dfsbcq 3333 . . . . . . 7  |-  ( w  =  Y  ->  ( [. w  /  y ]. ph  <->  [. Y  /  y ]. ph ) )
4342sbcbidv 3390 . . . . . 6  |-  ( w  =  Y  ->  ( [. X  /  x ]. [. w  /  y ]. ph  <->  [. X  /  x ]. [. Y  /  y ]. ph ) )
4443elrab 3261 . . . . 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 2630 . . . . . . 7  |-  ( X  e.  D  ->  F/_ x C )
47 elmptrab.s2 . . . . . . 7  |-  ( x  =  X  ->  B  =  C )
4846, 47csbiegf 3459 . . . . . 6  |-  ( X  e.  D  ->  [_ X  /  x ]_ B  =  C )
4948eleq2d 2537 . . . . 5  |-  ( X  e.  D  ->  ( Y  e.  [_ X  /  x ]_ B  <->  Y  e.  C ) )
5049anbi1d 704 . . . 4  |-  ( X  e.  D  ->  (
( Y  e.  [_ X  /  x ]_ B  /\  [. X  /  x ]. [. Y  /  y ]. ph )  <->  ( Y  e.  C  /\  [. X  /  x ]. [. Y  /  y ]. ph )
) )
51 nfv 1683 . . . . . 6  |-  F/ x ps
52 nfv 1683 . . . . . 6  |-  F/ y ps
53 nfv 1683 . . . . . 6  |-  F/ x  Y  e.  C
54 elmptrab.s1 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ph  <->  ps )
)
5551, 52, 53, 54sbc2iegf 3406 . . . . 5  |-  ( ( X  e.  D  /\  Y  e.  C )  ->  ( [. X  /  x ]. [. Y  / 
y ]. ph  <->  ps )
)
5655pm5.32da 641 . . . 4  |-  ( X  e.  D  ->  (
( Y  e.  C  /\  [. X  /  x ]. [. Y  /  y ]. ph )  <->  ( Y  e.  C  /\  ps )
) )
5745, 50, 563bitrd 279 . . 3  |-  ( X  e.  D  ->  ( Y  e.  { w  e.  [_ X  /  x ]_ B  |  [. X  /  x ]. [. w  /  y ]. ph }  <->  ( Y  e.  C  /\  ps ) ) )
58 3anass 977 . . . 4  |-  ( ( X  e.  D  /\  Y  e.  C  /\  ps )  <->  ( X  e.  D  /\  ( Y  e.  C  /\  ps ) ) )
5958baibr 902 . . 3  |-  ( X  e.  D  ->  (
( Y  e.  C  /\  ps )  <->  ( X  e.  D  /\  Y  e.  C  /\  ps )
) )
6041, 57, 593bitrd 279 . 2  |-  ( X  e.  D  ->  ( Y  e.  ( F `  X )  <->  ( X  e.  D  /\  Y  e.  C  /\  ps )
) )
614, 5, 60pm5.21nii 353 1  |-  ( Y  e.  ( F `  X )  <->  ( X  e.  D  /\  Y  e.  C  /\  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113   [.wsbc 3331   [_csb 3435    |-> cmpt 4505   dom cdm 4999   ` cfv 5588
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 5551  df-fun 5590  df-fv 5596
This theorem is referenced by:  elmptrab2  20092  isfbas  20093
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