Theorem ralrnmpt 5700

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 5380	. . . 4 |- ( A. x e. A B e. V -> F Fn A )
3		dfsbcq 2987	. . . . 5 |- ( w = ( F ` z ) -> ( [. w / y ]. ps <-> [. ( F ` z ) / y ]. ps ) )
4	3	ralrn 5696	. . . 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 1539	. . . . 5 |- F/ w ps
7		nfsbc1v 3004	. . . . 5 |- F/ y [. w / y ]. ps
8		sbceq1a 2995	. . . . 5 |- ( y = w -> ( ps <-> [. w / y ]. ps ) )
9	6, 7, 8	cbvral 2722	. . . 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 4122	. . . . . . 7 |- F/_ x ( x e. A |-> B )
12	1, 11	nfcxfr 2333	. . . . . 6 |- F/_ x F
13		nfcv 2336	. . . . . 6 |- F/_ x z
14	12, 13	nffv 5564	. . . . 5 |- F/_ x ( F ` z )
15		nfv 1539	. . . . 5 |- F/ x ps
16	14, 15	nfsbc 3006	. . . 4 |- F/ x [. ( F ` z ) / y ]. ps
17		nfv 1539	. . . 4 |- F/ z [. ( F ` x ) / y ]. ps
18		fveq2 5554	. . . . 5 |- ( z = x -> ( F ` z ) = ( F ` x ) )
19		dfsbcq 2987	. . . . 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 2722	. . 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 5641	. . . . . 6 |- ( ( x e. A /\ B e. V ) -> ( F ` x ) = B )
24		dfsbcq 2987	. . . . . 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 3018	. . . . . 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 2556	. . 3 |- ( A. x e. A B e. V -> A. x e. A ( [. ( F ` x ) / y ]. ps <-> ch ) )
31		ralbi 2626	. . 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 1364   e. wcel 2164   A.wral 2472   [.wsbc 2985   |-> cmpt 4090   ran crn 4660   Fn wfn 5249   ` cfv 5254

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262

This theorem is referenced by: (None)