Theorem fsovd 38302

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. (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 )

Assertion

Ref	Expression
fsovd	|- ( ph -> ( A O B ) = ( f e. ( ~P B ^m A ) |-> ( y e. B |-> { x e. A | y e. ( f ` x ) } ) ) )

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

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

Proof of Theorem fsovd

Step	Hyp	Ref	Expression
1		fsovd.fs	. . 3 |- O = ( a e. _V , b e. _V |-> ( f e. ( ~P b ^m a ) |-> ( y e. b |-> { x e. a | y e. ( f ` x ) } ) ) )
2	1	a1i 11	. 2 |- ( ph -> O = ( a e. _V , b e. _V |-> ( f e. ( ~P b ^m a ) |-> ( y e. b |-> { x e. a | y e. ( f ` x ) } ) ) ) )
3		pweq 4161	. . . . . 6 |- ( b = B -> ~P b = ~P B )
4	3	adantl 482	. . . . 5 |- ( ( a = A /\ b = B ) -> ~P b = ~P B )
5		simpl 473	. . . . 5 |- ( ( a = A /\ b = B ) -> a = A )
6	4, 5	oveq12d 6668	. . . 4 |- ( ( a = A /\ b = B ) -> ( ~P b ^m a ) = ( ~P B ^m A ) )
7		simpr 477	. . . . 5 |- ( ( a = A /\ b = B ) -> b = B )
8		rabeq 3192	. . . . . 6 |- ( a = A -> { x e. a | y e. ( f ` x ) } = { x e. A | y e. ( f ` x ) } )
9	8	adantr 481	. . . . 5 |- ( ( a = A /\ b = B ) -> { x e. a | y e. ( f ` x ) } = { x e. A | y e. ( f ` x ) } )
10	7, 9	mpteq12dv 4733	. . . 4 |- ( ( a = A /\ b = B ) -> ( y e. b |-> { x e. a | y e. ( f ` x ) } ) = ( y e. B |-> { x e. A | y e. ( f ` x ) } ) )
11	6, 10	mpteq12dv 4733	. . 3 |- ( ( a = A /\ b = B ) -> ( f e. ( ~P b ^m a ) |-> ( y e. b |-> { x e. a | y e. ( f ` x ) } ) ) = ( f e. ( ~P B ^m A ) |-> ( y e. B |-> { x e. A | y e. ( f ` x ) } ) ) )
12	11	adantl 482	. 2 |- ( ( ph /\ ( a = A /\ b = B ) ) -> ( f e. ( ~P b ^m a ) |-> ( y e. b |-> { x e. a | y e. ( f ` x ) } ) ) = ( f e. ( ~P B ^m A ) |-> ( y e. B |-> { x e. A | y e. ( f ` x ) } ) ) )
13		fsovd.a	. . 3 |- ( ph -> A e. V )
14	13	elexd 3214	. 2 |- ( ph -> A e. _V )
15		fsovd.b	. . 3 |- ( ph -> B e. W )
16	15	elexd 3214	. 2 |- ( ph -> B e. _V )
17		ovex 6678	. . . 4 |- ( ~P B ^m A ) e. _V
18	17	mptex 6486	. . 3 |- ( f e. ( ~P B ^m A ) |-> ( y e. B |-> { x e. A | y e. ( f ` x ) } ) ) e. _V
19	18	a1i 11	. 2 |- ( ph -> ( f e. ( ~P B ^m A ) |-> ( y e. B |-> { x e. A | y e. ( f ` x ) } ) ) e. _V )
20	2, 12, 14, 16, 19	ovmpt2d 6788	1 |- ( ph -> ( A O B ) = ( f e. ( ~P B ^m A ) |-> ( y e. B |-> { x e. A | y e. ( f ` x ) } ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = 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:  fsovrfovd  38303  fsovfvd  38304  fsovfd  38306  fsovcnvlem  38307