Theorem elmptrab 21630

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.

Step	Hyp	Ref	Expression
1		elmptrab.f	. . 3 |- F = ( x e. D |-> { y e. B | ph } )
2	1	mptrcl 6289	. 2 |- ( Y e. ( F ` X ) -> X e. D )
3		simp1 1061	. 2 |- ( ( X e. D /\ Y e. C /\ ps ) -> X e. D )
4		csbeq1 3536	. . . . . 6 |- ( z = X -> [_ z / x ]_ B = [_ X / x ]_ B )
5		dfsbcq 3437	. . . . . 6 |- ( z = X -> ( [. z / x ]. [. w / y ]. ph <-> [. X / x ]. [. w / y ]. ph ) )
6	4, 5	rabeqbidv 3195	. . . . 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 2764	. . . . . . 7 |- F/_ z { y e. B | ph }
8		nfsbc1v 3455	. . . . . . . 8 |- F/ x [. z / x ]. [. w / y ]. ph
9		nfcsb1v 3549	. . . . . . . 8 |- F/_ x [_ z / x ]_ B
10	8, 9	nfrab 3123	. . . . . . 7 |- F/_ x { w e. [_ z / x ]_ B | [. z / x ]. [. w / y ]. ph }
11		csbeq1a 3542	. . . . . . . . 9 |- ( x = z -> B = [_ z / x ]_ B )
12		sbceq1a 3446	. . . . . . . . 9 |- ( x = z -> ( ph <-> [. z / x ]. ph ) )
13	11, 12	rabeqbidv 3195	. . . . . . . 8 |- ( x = z -> { y e. B | ph } = { y e. [_ z / x ]_ B | [. z / x ]. ph } )
14		nfcv 2764	. . . . . . . . 9 |- F/_ w [_ z / x ]_ B
15		nfcv 2764	. . . . . . . . 9 |- F/_ y [_ z / x ]_ B
16		nfcv 2764	. . . . . . . . . 10 |- F/_ y z
17		nfsbc1v 3455	. . . . . . . . . 10 |- F/ y [. w / y ]. ph
18	16, 17	nfsbc 3457	. . . . . . . . 9 |- F/ y [. z / x ]. [. w / y ]. ph
19		nfv 1843	. . . . . . . . 9 |- F/ w [. z / x ]. ph
20		sbceq1a 3446	. . . . . . . . . . 11 |- ( y = w -> ( [. z / x ]. ph <-> [. w / y ]. [. z / x ]. ph ) )
21	20	equcoms 1947	. . . . . . . . . 10 |- ( w = y -> ( [. z / x ]. ph <-> [. w / y ]. [. z / x ]. ph ) )
22		sbccom 3509	. . . . . . . . . 10 |- ( [. z / x ]. [. w / y ]. ph <-> [. w / y ]. [. z / x ]. ph )
23	21, 22	syl6rbbr 279	. . . . . . . . 9 |- ( w = y -> ( [. z / x ]. [. w / y ]. ph <-> [. z / x ]. ph ) )
24	14, 15, 18, 19, 23	cbvrab 3198	. . . . . . . 8 |- { w e. [_ z / x ]_ B | [. z / x ]. [. w / y ]. ph } = { y e. [_ z / x ]_ B | [. z / x ]. ph }
25	13, 24	syl6eqr 2674	. . . . . . 7 |- ( x = z -> { y e. B | ph } = { w e. [_ z / x ]_ B | [. z / x ]. [. w / y ]. ph } )
26	7, 10, 25	cbvmpt 4749	. . . . . 6 |- ( x e. D |-> { y e. B | ph } ) = ( z e. D |-> { w e. [_ z / x ]_ B | [. z / x ]. [. w / y ]. ph } )
27	1, 26	eqtri 2644	. . . . 5 |- F = ( z e. D |-> { w e. [_ z / x ]_ B | [. z / x ]. [. w / y ]. ph } )
28		nfv 1843	. . . . . . . 8 |- F/ x z e. D
29	9	nfel1 2779	. . . . . . . 8 |- F/ x [_ z / x ]_ B e. V
30	28, 29	nfim 1825	. . . . . . 7 |- F/ x ( z e. D -> [_ z / x ]_ B e. V )
31		eleq1 2689	. . . . . . . 8 |- ( x = z -> ( x e. D <-> z e. D ) )
32	11	eleq1d 2686	. . . . . . . 8 |- ( x = z -> ( B e. V <-> [_ z / x ]_ B e. V ) )
33	31, 32	imbi12d 334	. . . . . . 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 )
35	30, 33, 34	chvar 2262	. . . . . 6 |- ( z e. D -> [_ z / x ]_ B e. V )
36		rabexg 4812	. . . . . 6 |- ( [_ z / x ]_ B e. V -> { w e. [_ z / x ]_ B | [. z / x ]. [. w / y ]. ph } e. _V )
37	35, 36	syl 17	. . . . 5 |- ( z e. D -> { w e. [_ z / x ]_ B | [. z / x ]. [. w / y ]. ph } e. _V )
38	6, 27, 37	fvmpt3 6286	. . . 4 |- ( X e. D -> ( F ` X ) = { w e. [_ X / x ]_ B | [. X / x ]. [. w / y ]. ph } )
39	38	eleq2d 2687	. . 3 |- ( X e. D -> ( Y e. ( F ` X ) <-> Y e. { w e. [_ X / x ]_ B | [. X / x ]. [. w / y ]. ph } ) )
40		dfsbcq 3437	. . . . . . 7 |- ( w = Y -> ( [. w / y ]. ph <-> [. Y / y ]. ph ) )
41	40	sbcbidv 3490	. . . . . 6 |- ( w = Y -> ( [. X / x ]. [. w / y ]. ph <-> [. X / x ]. [. Y / y ]. ph ) )
42	41	elrab 3363	. . . . 5 |- ( Y e. { w e. [_ X / x ]_ B | [. X / x ]. [. w / y ]. ph } <-> ( Y e. [_ X / x ]_ B /\ [. X / x ]. [. Y / y ]. ph ) )
43	42	a1i 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 2765	. . . . . . 7 |- ( X e. D -> F/_ x C )
45		elmptrab.s2	. . . . . . 7 |- ( x = X -> B = C )
46	44, 45	csbiegf 3557	. . . . . 6 |- ( X e. D -> [_ X / x ]_ B = C )
47	46	eleq2d 2687	. . . . 5 |- ( X e. D -> ( Y e. [_ X / x ]_ B <-> Y e. C ) )
48	47	anbi1d 741	. . . 4 |- ( X e. D -> ( ( Y e. [_ X / x ]_ B /\ [. X / x ]. [. Y / y ]. ph ) <-> ( Y e. C /\ [. X / x ]. [. Y / y ]. ph ) ) )
49		nfv 1843	. . . . . 6 |- F/ x ps
50		nfv 1843	. . . . . 6 |- F/ y ps
51		nfv 1843	. . . . . 6 |- F/ x Y e. C
52		elmptrab.s1	. . . . . 6 |- ( ( x = X /\ y = Y ) -> ( ph <-> ps ) )
53	49, 50, 51, 52	sbc2iegf 3504	. . . . 5 |- ( ( X e. D /\ Y e. C ) -> ( [. X / x ]. [. Y / y ]. ph <-> ps ) )
54	53	pm5.32da 673	. . . 4 |- ( X e. D -> ( ( Y e. C /\ [. X / x ]. [. Y / y ]. ph ) <-> ( Y e. C /\ ps ) ) )
55	43, 48, 54	3bitrd 294	. . 3 |- ( X e. D -> ( Y e. { w e. [_ X / x ]_ B | [. X / x ]. [. w / y ]. ph } <-> ( Y e. C /\ ps ) ) )
56		3anass 1042	. . . 4 |- ( ( X e. D /\ Y e. C /\ ps ) <-> ( X e. D /\ ( Y e. C /\ ps ) ) )
57	56	baibr 945	. . 3 |- ( X e. D -> ( ( Y e. C /\ ps ) <-> ( X e. D /\ Y e. C /\ ps ) ) )
58	39, 55, 57	3bitrd 294	. 2 |- ( X e. D -> ( Y e. ( F ` X ) <-> ( X e. D /\ Y e. C /\ ps ) ) )
59	2, 3, 58	pm5.21nii 368	1 |- ( Y e. ( F ` X ) <-> ( X e. D /\ Y e. C /\ ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   [.wsbc 3435   [_csb 3533   |-> cmpt 4729   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  elmptrab2OLD  21631  elmptrab2  21632  isfbas  21633