Theorem fsovfvd 38304

Description: Value of the operator, ( A O B ), which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, A and B, when applied to function F. (Contributed by RP, 25-Apr-2021.)

Hypotheses

Ref	Expression
fsovd.fs	|- O = ( a e. _V , b e. _V |-> ( f e. ( ~P b ^m a ) |-> ( y e. b |-> { x e. a | y e. ( f ` x ) } ) ) )
fsovd.a	|- ( ph -> A e. V )
fsovd.b	|- ( ph -> B e. W )
fsovfvd.g	|- G = ( A O B )
fsovfvd.f	|- ( ph -> F e. ( ~P B ^m A ) )

Assertion

Ref	Expression
fsovfvd	|- ( ph -> ( G ` F ) = ( y e. B |-> { x e. A | y e. ( F ` x ) } ) )

Distinct variable groups:   A, a, b, f, x   y, A, a, b, f   B, a, b, f, y   f, F, x   y, F   ph, a, b, f

Allowed substitution hints:   ph( x, y)   B( x)   F( a, b)   G( x, y, f, a, b)   O( x, y, f, a, b)   V( x, y, f, a, b)   W( x, y, f, a, b)

Proof of Theorem fsovfvd

Step	Hyp	Ref	Expression
1		fsovfvd.g	. . 3 |- G = ( A O B )
2		fsovd.fs	. . . 4 |- O = ( a e. _V , b e. _V |-> ( f e. ( ~P b ^m a ) |-> ( y e. b |-> { x e. a | y e. ( f ` x ) } ) ) )
3		fsovd.a	. . . 4 |- ( ph -> A e. V )
4		fsovd.b	. . . 4 |- ( ph -> B e. W )
5	2, 3, 4	fsovd 38302	. . 3 |- ( ph -> ( A O B ) = ( f e. ( ~P B ^m A ) |-> ( y e. B |-> { x e. A | y e. ( f ` x ) } ) ) )
6	1, 5	syl5eq 2668	. 2 |- ( ph -> G = ( f e. ( ~P B ^m A ) |-> ( y e. B |-> { x e. A | y e. ( f ` x ) } ) ) )
7		fveq1 6190	. . . . . 6 |- ( f = F -> ( f ` x ) = ( F ` x ) )
8	7	eleq2d 2687	. . . . 5 |- ( f = F -> ( y e. ( f ` x ) <-> y e. ( F ` x ) ) )
9	8	rabbidv 3189	. . . 4 |- ( f = F -> { x e. A | y e. ( f ` x ) } = { x e. A | y e. ( F ` x ) } )
10	9	mpteq2dv 4745	. . 3 |- ( f = F -> ( y e. B |-> { x e. A | y e. ( f ` x ) } ) = ( y e. B |-> { x e. A | y e. ( F ` x ) } ) )
11	10	adantl 482	. 2 |- ( ( ph /\ f = F ) -> ( y e. B |-> { x e. A | y e. ( f ` x ) } ) = ( y e. B |-> { x e. A | y e. ( F ` x ) } ) )
12		fsovfvd.f	. 2 |- ( ph -> F e. ( ~P B ^m A ) )
13		mptexg 6484	. . 3 |- ( B e. W -> ( y e. B |-> { x e. A | y e. ( F ` x ) } ) e. _V )
14	4, 13	syl 17	. 2 |- ( ph -> ( y e. B |-> { x e. A | y e. ( F ` x ) } ) e. _V )
15	6, 11, 12, 14	fvmptd 6288	1 |- ( ph -> ( G ` F ) = ( y e. B |-> { x e. A | y e. ( F ` x ) } ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   ~Pcpw 4158   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  fsovfvfvd  38305  fsovcnvfvd  38309