Theorem ralrnmpt 5330

Description: A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)

Hypotheses

Ref	Expression
ralrnmpt.1	|- F = ( x e. A |-> B )
ralrnmpt.2	|- ( y = B -> ( ps <-> ch ) )

Assertion

Ref	Expression
ralrnmpt	|- ( A. x e. A B e. V -> ( A. y e. ran F ps <-> A. x e. A ch ) )

Distinct variable groups:   x, A   y, B   ch, y   y, F   ps, x

Allowed substitution hints:   ps( y)   ch( x)   A( y)   B( x)   F( x)   V( x, y)

Proof of Theorem ralrnmpt

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ralrnmpt.1	. . . . 5 |- F = ( x e. A |-> B )
2	1	fnmpt 5045	. . . 4 |- ( A. x e. A B e. V -> F Fn A )
3		dfsbcq 2817	. . . . 5 |- ( w = ( F ` z ) -> ( [. w / y ]. ps <-> [. ( F ` z ) / y ]. ps ) )
4	3	ralrn 5326	. . . 4 |- ( F Fn A -> ( A. w e. ran F [. w / y ]. ps <-> A. z e. A [. ( F ` z ) / y ]. ps ) )
5	2, 4	syl 14	. . 3 |- ( A. x e. A B e. V -> ( A. w e. ran F [. w / y ]. ps <-> A. z e. A [. ( F ` z ) / y ]. ps ) )
6		nfv 1461	. . . . 5 |- F/ w ps
7		nfsbc1v 2833	. . . . 5 |- F/ y [. w / y ]. ps
8		sbceq1a 2824	. . . . 5 |- ( y = w -> ( ps <-> [. w / y ]. ps ) )
9	6, 7, 8	cbvral 2573	. . . 4 |- ( A. y e. ran F ps <-> A. w e. ran F [. w / y ]. ps )
10	9	bicomi 130	. . 3 |- ( A. w e. ran F [. w / y ]. ps <-> A. y e. ran F ps )
11		nfmpt1 3871	. . . . . . 7 |- F/_ x ( x e. A |-> B )
12	1, 11	nfcxfr 2216	. . . . . 6 |- F/_ x F
13		nfcv 2219	. . . . . 6 |- F/_ x z
14	12, 13	nffv 5205	. . . . 5 |- F/_ x ( F ` z )
15		nfv 1461	. . . . 5 |- F/ x ps
16	14, 15	nfsbc 2835	. . . 4 |- F/ x [. ( F ` z ) / y ]. ps
17		nfv 1461	. . . 4 |- F/ z [. ( F ` x ) / y ]. ps
18		fveq2 5198	. . . . 5 |- ( z = x -> ( F ` z ) = ( F ` x ) )
19		dfsbcq 2817	. . . . 5 |- ( ( F ` z ) = ( F ` x ) -> ( [. ( F ` z ) / y ]. ps <-> [. ( F ` x ) / y ]. ps ) )
20	18, 19	syl 14	. . . 4 |- ( z = x -> ( [. ( F ` z ) / y ]. ps <-> [. ( F ` x ) / y ]. ps ) )
21	16, 17, 20	cbvral 2573	. . 3 |- ( A. z e. A [. ( F ` z ) / y ]. ps <-> A. x e. A [. ( F ` x ) / y ]. ps )
22	5, 10, 21	3bitr3g 220	. 2 |- ( A. x e. A B e. V -> ( A. y e. ran F ps <-> A. x e. A [. ( F ` x ) / y ]. ps ) )
23	1	fvmpt2 5275	. . . . . 6 |- ( ( x e. A /\ B e. V ) -> ( F ` x ) = B )
24		dfsbcq 2817	. . . . . 6 |- ( ( F ` x ) = B -> ( [. ( F ` x ) / y ]. ps <-> [. B / y ]. ps ) )
25	23, 24	syl 14	. . . . 5 |- ( ( x e. A /\ B e. V ) -> ( [. ( F ` x ) / y ]. ps <-> [. B / y ]. ps ) )
26		ralrnmpt.2	. . . . . . 7 |- ( y = B -> ( ps <-> ch ) )
27	26	sbcieg 2846	. . . . . 6 |- ( B e. V -> ( [. B / y ]. ps <-> ch ) )
28	27	adantl 271	. . . . 5 |- ( ( x e. A /\ B e. V ) -> ( [. B / y ]. ps <-> ch ) )
29	25, 28	bitrd 186	. . . 4 |- ( ( x e. A /\ B e. V ) -> ( [. ( F ` x ) / y ]. ps <-> ch ) )
30	29	ralimiaa 2425	. . 3 |- ( A. x e. A B e. V -> A. x e. A ( [. ( F ` x ) / y ]. ps <-> ch ) )
31		ralbi 2489	. . 3 |- ( A. x e. A ( [. ( F ` x ) / y ]. ps <-> ch ) -> ( A. x e. A [. ( F ` x ) / y ]. ps <-> A. x e. A ch ) )
32	30, 31	syl 14	. 2 |- ( A. x e. A B e. V -> ( A. x e. A [. ( F ` x ) / y ]. ps <-> A. x e. A ch ) )
33	22, 32	bitrd 186	1 |- ( A. x e. A B e. V -> ( A. y e. ran F ps <-> A. x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348   [.wsbc 2815   |-> cmpt 3839   ran crn 4364   Fn wfn 4917   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930

This theorem is referenced by: (None)