Theorem mapsncnv 7904

Description: Expression for the inverse of the canonical map between a set and its set of singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)

Hypotheses

Ref	Expression
mapsncnv.s	|- S = { X }
mapsncnv.b	|- B e. _V
mapsncnv.x	|- X e. _V
mapsncnv.f	|- F = ( x e. ( B ^m S ) |-> ( x ` X ) )

Assertion

Ref	Expression
mapsncnv	|- `' F = ( y e. B |-> ( S X. { y } ) )

Distinct variable groups:   x, B, y   x, S, y   y, X

Allowed substitution hints:   F( x, y)   X( x)

Proof of Theorem mapsncnv

Step	Hyp	Ref	Expression
1		elmapi 7879	. . . . . . . . 9 |- ( x e. ( B ^m { X } ) -> x : { X } --> B )
2		mapsncnv.x	. . . . . . . . . 10 |- X e. _V
3	2	snid 4208	. . . . . . . . 9 |- X e. { X }
4		ffvelrn 6357	. . . . . . . . 9 |- ( ( x : { X } --> B /\ X e. { X } ) -> ( x ` X ) e. B )
5	1, 3, 4	sylancl 694	. . . . . . . 8 |- ( x e. ( B ^m { X } ) -> ( x ` X ) e. B )
6		eqid 2622	. . . . . . . . 9 |- { X } = { X }
7		mapsncnv.b	. . . . . . . . 9 |- B e. _V
8	6, 7, 2	mapsnconst 7903	. . . . . . . 8 |- ( x e. ( B ^m { X } ) -> x = ( { X } X. { ( x ` X ) } ) )
9	5, 8	jca 554	. . . . . . 7 |- ( x e. ( B ^m { X } ) -> ( ( x ` X ) e. B /\ x = ( { X } X. { ( x ` X ) } ) ) )
10		eleq1 2689	. . . . . . . 8 |- ( y = ( x ` X ) -> ( y e. B <-> ( x ` X ) e. B ) )
11		sneq 4187	. . . . . . . . . 10 |- ( y = ( x ` X ) -> { y } = { ( x ` X ) } )
12	11	xpeq2d 5139	. . . . . . . . 9 |- ( y = ( x ` X ) -> ( { X } X. { y } ) = ( { X } X. { ( x ` X ) } ) )
13	12	eqeq2d 2632	. . . . . . . 8 |- ( y = ( x ` X ) -> ( x = ( { X } X. { y } ) <-> x = ( { X } X. { ( x ` X ) } ) ) )
14	10, 13	anbi12d 747	. . . . . . 7 |- ( y = ( x ` X ) -> ( ( y e. B /\ x = ( { X } X. { y } ) ) <-> ( ( x ` X ) e. B /\ x = ( { X } X. { ( x ` X ) } ) ) ) )
15	9, 14	syl5ibrcom 237	. . . . . 6 |- ( x e. ( B ^m { X } ) -> ( y = ( x ` X ) -> ( y e. B /\ x = ( { X } X. { y } ) ) ) )
16	15	imp 445	. . . . 5 |- ( ( x e. ( B ^m { X } ) /\ y = ( x ` X ) ) -> ( y e. B /\ x = ( { X } X. { y } ) ) )
17		fconst6g 6094	. . . . . . . . 9 |- ( y e. B -> ( { X } X. { y } ) : { X } --> B )
18		snex 4908	. . . . . . . . . 10 |- { X } e. _V
19	7, 18	elmap 7886	. . . . . . . . 9 |- ( ( { X } X. { y } ) e. ( B ^m { X } ) <-> ( { X } X. { y } ) : { X } --> B )
20	17, 19	sylibr 224	. . . . . . . 8 |- ( y e. B -> ( { X } X. { y } ) e. ( B ^m { X } ) )
21		vex 3203	. . . . . . . . . . 11 |- y e. _V
22	21	fvconst2 6469	. . . . . . . . . 10 |- ( X e. { X } -> ( ( { X } X. { y } ) ` X ) = y )
23	3, 22	mp1i 13	. . . . . . . . 9 |- ( y e. B -> ( ( { X } X. { y } ) ` X ) = y )
24	23	eqcomd 2628	. . . . . . . 8 |- ( y e. B -> y = ( ( { X } X. { y } ) ` X ) )
25	20, 24	jca 554	. . . . . . 7 |- ( y e. B -> ( ( { X } X. { y } ) e. ( B ^m { X } ) /\ y = ( ( { X } X. { y } ) ` X ) ) )
26		eleq1 2689	. . . . . . . 8 |- ( x = ( { X } X. { y } ) -> ( x e. ( B ^m { X } ) <-> ( { X } X. { y } ) e. ( B ^m { X } ) ) )
27		fveq1 6190	. . . . . . . . 9 |- ( x = ( { X } X. { y } ) -> ( x ` X ) = ( ( { X } X. { y } ) ` X ) )
28	27	eqeq2d 2632	. . . . . . . 8 |- ( x = ( { X } X. { y } ) -> ( y = ( x ` X ) <-> y = ( ( { X } X. { y } ) ` X ) ) )
29	26, 28	anbi12d 747	. . . . . . 7 |- ( x = ( { X } X. { y } ) -> ( ( x e. ( B ^m { X } ) /\ y = ( x ` X ) ) <-> ( ( { X } X. { y } ) e. ( B ^m { X } ) /\ y = ( ( { X } X. { y } ) ` X ) ) ) )
30	25, 29	syl5ibrcom 237	. . . . . 6 |- ( y e. B -> ( x = ( { X } X. { y } ) -> ( x e. ( B ^m { X } ) /\ y = ( x ` X ) ) ) )
31	30	imp 445	. . . . 5 |- ( ( y e. B /\ x = ( { X } X. { y } ) ) -> ( x e. ( B ^m { X } ) /\ y = ( x ` X ) ) )
32	16, 31	impbii 199	. . . 4 |- ( ( x e. ( B ^m { X } ) /\ y = ( x ` X ) ) <-> ( y e. B /\ x = ( { X } X. { y } ) ) )
33		mapsncnv.s	. . . . . . 7 |- S = { X }
34	33	oveq2i 6661	. . . . . 6 |- ( B ^m S ) = ( B ^m { X } )
35	34	eleq2i 2693	. . . . 5 |- ( x e. ( B ^m S ) <-> x e. ( B ^m { X } ) )
36	35	anbi1i 731	. . . 4 |- ( ( x e. ( B ^m S ) /\ y = ( x ` X ) ) <-> ( x e. ( B ^m { X } ) /\ y = ( x ` X ) ) )
37	33	xpeq1i 5135	. . . . . 6 |- ( S X. { y } ) = ( { X } X. { y } )
38	37	eqeq2i 2634	. . . . 5 |- ( x = ( S X. { y } ) <-> x = ( { X } X. { y } ) )
39	38	anbi2i 730	. . . 4 |- ( ( y e. B /\ x = ( S X. { y } ) ) <-> ( y e. B /\ x = ( { X } X. { y } ) ) )
40	32, 36, 39	3bitr4i 292	. . 3 |- ( ( x e. ( B ^m S ) /\ y = ( x ` X ) ) <-> ( y e. B /\ x = ( S X. { y } ) ) )
41	40	opabbii 4717	. 2 |- { <. y , x >. | ( x e. ( B ^m S ) /\ y = ( x ` X ) ) } = { <. y , x >. | ( y e. B /\ x = ( S X. { y } ) ) }
42		mapsncnv.f	. . . . 5 |- F = ( x e. ( B ^m S ) |-> ( x ` X ) )
43		df-mpt 4730	. . . . 5 |- ( x e. ( B ^m S ) |-> ( x ` X ) ) = { <. x , y >. | ( x e. ( B ^m S ) /\ y = ( x ` X ) ) }
44	42, 43	eqtri 2644	. . . 4 |- F = { <. x , y >. | ( x e. ( B ^m S ) /\ y = ( x ` X ) ) }
45	44	cnveqi 5297	. . 3 |- `' F = `' { <. x , y >. | ( x e. ( B ^m S ) /\ y = ( x ` X ) ) }
46		cnvopab 5533	. . 3 |- `' { <. x , y >. | ( x e. ( B ^m S ) /\ y = ( x ` X ) ) } = { <. y , x >. | ( x e. ( B ^m S ) /\ y = ( x ` X ) ) }
47	45, 46	eqtri 2644	. 2 |- `' F = { <. y , x >. | ( x e. ( B ^m S ) /\ y = ( x ` X ) ) }
48		df-mpt 4730	. 2 |- ( y e. B |-> ( S X. { y } ) ) = { <. y , x >. | ( y e. B /\ x = ( S X. { y } ) ) }
49	41, 47, 48	3eqtr4i 2654	1 |- `' F = ( y e. B |-> ( S X. { y } ) )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   {copab 4712   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by:  mapsnf1o2  7905  mapsnf1o3  7906  coe1sfi  19583  evl1var  19700  pf1mpf  19716  pf1ind  19719  deg1val  23856