Theorem ralrnmpt 5455

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 5153	. . . 4 |- ( A. x e. A B e. V -> F Fn A )
3		dfsbcq 2843	. . . . 5 |- ( w = ( F ` z ) -> ( [. w / y ]. ps <-> [. ( F ` z ) / y ]. ps ) )
4	3	ralrn 5451	. . . 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 1467	. . . . 5 |- F/ w ps
7		nfsbc1v 2859	. . . . 5 |- F/ y [. w / y ]. ps
8		sbceq1a 2850	. . . . 5 |- ( y = w -> ( ps <-> [. w / y ]. ps ) )
9	6, 7, 8	cbvral 2587	. . . 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 3937	. . . . . . 7 |- F/_ x ( x e. A |-> B )
12	1, 11	nfcxfr 2226	. . . . . 6 |- F/_ x F
13		nfcv 2229	. . . . . 6 |- F/_ x z
14	12, 13	nffv 5328	. . . . 5 |- F/_ x ( F ` z )
15		nfv 1467	. . . . 5 |- F/ x ps
16	14, 15	nfsbc 2861	. . . 4 |- F/ x [. ( F ` z ) / y ]. ps
17		nfv 1467	. . . 4 |- F/ z [. ( F ` x ) / y ]. ps
18		fveq2 5318	. . . . 5 |- ( z = x -> ( F ` z ) = ( F ` x ) )
19		dfsbcq 2843	. . . . 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 2587	. . 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 5399	. . . . . 6 |- ( ( x e. A /\ B e. V ) -> ( F ` x ) = B )
24		dfsbcq 2843	. . . . . 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 2872	. . . . . 6 |- ( B e. V -> ( [. B / y ]. ps <-> ch ) )
28	27	adantl 272	. . . . 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 2438	. . 3 |- ( A. x e. A B e. V -> A. x e. A ( [. ( F ` x ) / y ]. ps <-> ch ) )
31		ralbi 2502	. . 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 1290   e. wcel 1439   A.wral 2360   [.wsbc 2841   |-> cmpt 3905   ran crn 4453   Fn wfn 5023   ` cfv 5028

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-csb 2935  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-iota 4993  df-fun 5030  df-fn 5031  df-fv 5036

This theorem is referenced by: (None)