Theorem ralrnmpt 5612

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 5299	. . . 4 |- ( A. x e. A B e. V -> F Fn A )
3		dfsbcq 2939	. . . . 5 |- ( w = ( F ` z ) -> ( [. w / y ]. ps <-> [. ( F ` z ) / y ]. ps ) )
4	3	ralrn 5608	. . . 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 1508	. . . . 5 |- F/ w ps
7		nfsbc1v 2955	. . . . 5 |- F/ y [. w / y ]. ps
8		sbceq1a 2946	. . . . 5 |- ( y = w -> ( ps <-> [. w / y ]. ps ) )
9	6, 7, 8	cbvral 2676	. . . 4 |- ( A. y e. ran F ps <-> A. w e. ran F [. w / y ]. ps )
10	9	bicomi 131	. . 3 |- ( A. w e. ran F [. w / y ]. ps <-> A. y e. ran F ps )
11		nfmpt1 4060	. . . . . . 7 |- F/_ x ( x e. A |-> B )
12	1, 11	nfcxfr 2296	. . . . . 6 |- F/_ x F
13		nfcv 2299	. . . . . 6 |- F/_ x z
14	12, 13	nffv 5481	. . . . 5 |- F/_ x ( F ` z )
15		nfv 1508	. . . . 5 |- F/ x ps
16	14, 15	nfsbc 2957	. . . 4 |- F/ x [. ( F ` z ) / y ]. ps
17		nfv 1508	. . . 4 |- F/ z [. ( F ` x ) / y ]. ps
18		fveq2 5471	. . . . 5 |- ( z = x -> ( F ` z ) = ( F ` x ) )
19		dfsbcq 2939	. . . . 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 2676	. . 3 |- ( A. z e. A [. ( F ` z ) / y ]. ps <-> A. x e. A [. ( F ` x ) / y ]. ps )
22	5, 10, 21	3bitr3g 221	. 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 5554	. . . . . 6 |- ( ( x e. A /\ B e. V ) -> ( F ` x ) = B )
24		dfsbcq 2939	. . . . . 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 2969	. . . . . 6 |- ( B e. V -> ( [. B / y ]. ps <-> ch ) )
28	27	adantl 275	. . . . 5 |- ( ( x e. A /\ B e. V ) -> ( [. B / y ]. ps <-> ch ) )
29	25, 28	bitrd 187	. . . 4 |- ( ( x e. A /\ B e. V ) -> ( [. ( F ` x ) / y ]. ps <-> ch ) )
30	29	ralimiaa 2519	. . 3 |- ( A. x e. A B e. V -> A. x e. A ( [. ( F ` x ) / y ]. ps <-> ch ) )
31		ralbi 2589	. . 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 187	1 |- ( A. x e. A B e. V -> ( A. y e. ran F ps <-> A. x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   A.wral 2435   [.wsbc 2937   |-> cmpt 4028   ran crn 4590   Fn wfn 5168   ` cfv 5173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4085  ax-pow 4138  ax-pr 4172

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4029  df-mpt 4030  df-id 4256  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-iota 5138  df-fun 5175  df-fn 5176  df-fv 5181

This theorem is referenced by: (None)