Theorem ralrnmpt 5657

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 5341	. . . 4 |- ( A. x e. A B e. V -> F Fn A )
3		dfsbcq 2964	. . . . 5 |- ( w = ( F ` z ) -> ( [. w / y ]. ps <-> [. ( F ` z ) / y ]. ps ) )
4	3	ralrn 5653	. . . 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 1528	. . . . 5 |- F/ w ps
7		nfsbc1v 2981	. . . . 5 |- F/ y [. w / y ]. ps
8		sbceq1a 2972	. . . . 5 |- ( y = w -> ( ps <-> [. w / y ]. ps ) )
9	6, 7, 8	cbvral 2699	. . . 4 |- ( A. y e. ran F ps <-> A. w e. ran F [. w / y ]. ps )
10	9	bicomi 132	. . 3 |- ( A. w e. ran F [. w / y ]. ps <-> A. y e. ran F ps )
11		nfmpt1 4095	. . . . . . 7 |- F/_ x ( x e. A |-> B )
12	1, 11	nfcxfr 2316	. . . . . 6 |- F/_ x F
13		nfcv 2319	. . . . . 6 |- F/_ x z
14	12, 13	nffv 5524	. . . . 5 |- F/_ x ( F ` z )
15		nfv 1528	. . . . 5 |- F/ x ps
16	14, 15	nfsbc 2983	. . . 4 |- F/ x [. ( F ` z ) / y ]. ps
17		nfv 1528	. . . 4 |- F/ z [. ( F ` x ) / y ]. ps
18		fveq2 5514	. . . . 5 |- ( z = x -> ( F ` z ) = ( F ` x ) )
19		dfsbcq 2964	. . . . 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 2699	. . 3 |- ( A. z e. A [. ( F ` z ) / y ]. ps <-> A. x e. A [. ( F ` x ) / y ]. ps )
22	5, 10, 21	3bitr3g 222	. 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 5598	. . . . . 6 |- ( ( x e. A /\ B e. V ) -> ( F ` x ) = B )
24		dfsbcq 2964	. . . . . 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 2995	. . . . . 6 |- ( B e. V -> ( [. B / y ]. ps <-> ch ) )
28	27	adantl 277	. . . . 5 |- ( ( x e. A /\ B e. V ) -> ( [. B / y ]. ps <-> ch ) )
29	25, 28	bitrd 188	. . . 4 |- ( ( x e. A /\ B e. V ) -> ( [. ( F ` x ) / y ]. ps <-> ch ) )
30	29	ralimiaa 2539	. . 3 |- ( A. x e. A B e. V -> A. x e. A ( [. ( F ` x ) / y ]. ps <-> ch ) )
31		ralbi 2609	. . 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 188	1 |- ( A. x e. A B e. V -> ( A. y e. ran F ps <-> A. x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   A.wral 2455   [.wsbc 2962   |-> cmpt 4063   ran crn 4626   Fn wfn 5210   ` cfv 5215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223

This theorem is referenced by: (None)