Theorem ssenen 7032

Description: Equinumerosity of equinumerous subsets of a set. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)

Assertion

Ref	Expression
ssenen	|- ( A ~~ B -> { x | ( x C_ A /\ x ~~ C ) } ~~ { x | ( x C_ B /\ x ~~ C ) } )

Distinct variable groups:   x, A   x, B   x, C

Proof of Theorem ssenen

Dummy variables y z f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 6912	. . 3 |- ( A ~~ B <-> E. f f : A -1-1-onto-> B )
2		f1odm 5584	. . . . . . 7 |- ( f : A -1-1-onto-> B -> dom f = A )
3		vex 2803	. . . . . . . 8 |- f e. _V
4	3	dmex 4997	. . . . . . 7 |- dom f e. _V
5	2, 4	eqeltrrdi 2321	. . . . . 6 |- ( f : A -1-1-onto-> B -> A e. _V )
6		pwexg 4268	. . . . . 6 |- ( A e. _V -> ~P A e. _V )
7		inex1g 4223	. . . . . 6 |- ( ~P A e. _V -> ( ~P A i^i { x | x ~~ C } ) e. _V )
8	5, 6, 7	3syl 17	. . . . 5 |- ( f : A -1-1-onto-> B -> ( ~P A i^i { x | x ~~ C } ) e. _V )
9		f1ofo 5587	. . . . . . . 8 |- ( f : A -1-1-onto-> B -> f : A -onto-> B )
10		forn 5559	. . . . . . . 8 |- ( f : A -onto-> B -> ran f = B )
11	9, 10	syl 14	. . . . . . 7 |- ( f : A -1-1-onto-> B -> ran f = B )
12	3	rnex 4998	. . . . . . 7 |- ran f e. _V
13	11, 12	eqeltrrdi 2321	. . . . . 6 |- ( f : A -1-1-onto-> B -> B e. _V )
14		pwexg 4268	. . . . . 6 |- ( B e. _V -> ~P B e. _V )
15		inex1g 4223	. . . . . 6 |- ( ~P B e. _V -> ( ~P B i^i { x | x ~~ C } ) e. _V )
16	13, 14, 15	3syl 17	. . . . 5 |- ( f : A -1-1-onto-> B -> ( ~P B i^i { x | x ~~ C } ) e. _V )
17		f1of1 5579	. . . . . . . . . . 11 |- ( f : A -1-1-onto-> B -> f : A -1-1-> B )
18	17	adantr 276	. . . . . . . . . 10 |- ( ( f : A -1-1-onto-> B /\ y C_ A ) -> f : A -1-1-> B )
19	13	adantr 276	. . . . . . . . . 10 |- ( ( f : A -1-1-onto-> B /\ y C_ A ) -> B e. _V )
20		simpr 110	. . . . . . . . . 10 |- ( ( f : A -1-1-onto-> B /\ y C_ A ) -> y C_ A )
21		vex 2803	. . . . . . . . . . 11 |- y e. _V
22	21	a1i 9	. . . . . . . . . 10 |- ( ( f : A -1-1-onto-> B /\ y C_ A ) -> y e. _V )
23		f1imaen2g 6962	. . . . . . . . . 10 |- ( ( ( f : A -1-1-> B /\ B e. _V ) /\ ( y C_ A /\ y e. _V ) ) -> ( f " y ) ~~ y )
24	18, 19, 20, 22, 23	syl22anc 1272	. . . . . . . . 9 |- ( ( f : A -1-1-onto-> B /\ y C_ A ) -> ( f " y ) ~~ y )
25		entr 6953	. . . . . . . . 9 |- ( ( ( f " y ) ~~ y /\ y ~~ C ) -> ( f " y ) ~~ C )
26	24, 25	sylan 283	. . . . . . . 8 |- ( ( ( f : A -1-1-onto-> B /\ y C_ A ) /\ y ~~ C ) -> ( f " y ) ~~ C )
27	26	expl 378	. . . . . . 7 |- ( f : A -1-1-onto-> B -> ( ( y C_ A /\ y ~~ C ) -> ( f " y ) ~~ C ) )
28		imassrn 5085	. . . . . . . . 9 |- ( f " y ) C_ ran f
29	28, 10	sseqtrid 3275	. . . . . . . 8 |- ( f : A -onto-> B -> ( f " y ) C_ B )
30	9, 29	syl 14	. . . . . . 7 |- ( f : A -1-1-onto-> B -> ( f " y ) C_ B )
31	27, 30	jctild 316	. . . . . 6 |- ( f : A -1-1-onto-> B -> ( ( y C_ A /\ y ~~ C ) -> ( ( f " y ) C_ B /\ ( f " y ) ~~ C ) ) )
32		elin 3388	. . . . . . 7 |- ( y e. ( ~P A i^i { x | x ~~ C } ) <-> ( y e. ~P A /\ y e. { x | x ~~ C } ) )
33	21	elpw 3656	. . . . . . . 8 |- ( y e. ~P A <-> y C_ A )
34		breq1 4089	. . . . . . . . 9 |- ( x = y -> ( x ~~ C <-> y ~~ C ) )
35	21, 34	elab 2948	. . . . . . . 8 |- ( y e. { x | x ~~ C } <-> y ~~ C )
36	33, 35	anbi12i 460	. . . . . . 7 |- ( ( y e. ~P A /\ y e. { x | x ~~ C } ) <-> ( y C_ A /\ y ~~ C ) )
37	32, 36	bitri 184	. . . . . 6 |- ( y e. ( ~P A i^i { x | x ~~ C } ) <-> ( y C_ A /\ y ~~ C ) )
38		elin 3388	. . . . . . 7 |- ( ( f " y ) e. ( ~P B i^i { x | x ~~ C } ) <-> ( ( f " y ) e. ~P B /\ ( f " y ) e. { x | x ~~ C } ) )
39	3	imaex 5089	. . . . . . . . 9 |- ( f " y ) e. _V
40	39	elpw 3656	. . . . . . . 8 |- ( ( f " y ) e. ~P B <-> ( f " y ) C_ B )
41		breq1 4089	. . . . . . . . 9 |- ( x = ( f " y ) -> ( x ~~ C <-> ( f " y ) ~~ C ) )
42	39, 41	elab 2948	. . . . . . . 8 |- ( ( f " y ) e. { x | x ~~ C } <-> ( f " y ) ~~ C )
43	40, 42	anbi12i 460	. . . . . . 7 |- ( ( ( f " y ) e. ~P B /\ ( f " y ) e. { x | x ~~ C } ) <-> ( ( f " y ) C_ B /\ ( f " y ) ~~ C ) )
44	38, 43	bitri 184	. . . . . 6 |- ( ( f " y ) e. ( ~P B i^i { x | x ~~ C } ) <-> ( ( f " y ) C_ B /\ ( f " y ) ~~ C ) )
45	31, 37, 44	3imtr4g 205	. . . . 5 |- ( f : A -1-1-onto-> B -> ( y e. ( ~P A i^i { x | x ~~ C } ) -> ( f " y ) e. ( ~P B i^i { x | x ~~ C } ) ) )
46		f1ocnv 5593	. . . . . . 7 |- ( f : A -1-1-onto-> B -> `' f : B -1-1-onto-> A )
47		f1of1 5579	. . . . . . . . . . . 12 |- ( `' f : B -1-1-onto-> A -> `' f : B -1-1-> A )
48		f1f1orn 5591	. . . . . . . . . . . 12 |- ( `' f : B -1-1-> A -> `' f : B -1-1-onto-> ran `' f )
49		f1of1 5579	. . . . . . . . . . . 12 |- ( `' f : B -1-1-onto-> ran `' f -> `' f : B -1-1-> ran `' f )
50	47, 48, 49	3syl 17	. . . . . . . . . . 11 |- ( `' f : B -1-1-onto-> A -> `' f : B -1-1-> ran `' f )
51		vex 2803	. . . . . . . . . . . 12 |- z e. _V
52	51	f1imaen 6963	. . . . . . . . . . 11 |- ( ( `' f : B -1-1-> ran `' f /\ z C_ B ) -> ( `' f " z ) ~~ z )
53	50, 52	sylan 283	. . . . . . . . . 10 |- ( ( `' f : B -1-1-onto-> A /\ z C_ B ) -> ( `' f " z ) ~~ z )
54		entr 6953	. . . . . . . . . 10 |- ( ( ( `' f " z ) ~~ z /\ z ~~ C ) -> ( `' f " z ) ~~ C )
55	53, 54	sylan 283	. . . . . . . . 9 |- ( ( ( `' f : B -1-1-onto-> A /\ z C_ B ) /\ z ~~ C ) -> ( `' f " z ) ~~ C )
56	55	expl 378	. . . . . . . 8 |- ( `' f : B -1-1-onto-> A -> ( ( z C_ B /\ z ~~ C ) -> ( `' f " z ) ~~ C ) )
57		f1ofo 5587	. . . . . . . . 9 |- ( `' f : B -1-1-onto-> A -> `' f : B -onto-> A )
58		imassrn 5085	. . . . . . . . . 10 |- ( `' f " z ) C_ ran `' f
59		forn 5559	. . . . . . . . . 10 |- ( `' f : B -onto-> A -> ran `' f = A )
60	58, 59	sseqtrid 3275	. . . . . . . . 9 |- ( `' f : B -onto-> A -> ( `' f " z ) C_ A )
61	57, 60	syl 14	. . . . . . . 8 |- ( `' f : B -1-1-onto-> A -> ( `' f " z ) C_ A )
62	56, 61	jctild 316	. . . . . . 7 |- ( `' f : B -1-1-onto-> A -> ( ( z C_ B /\ z ~~ C ) -> ( ( `' f " z ) C_ A /\ ( `' f " z ) ~~ C ) ) )
63	46, 62	syl 14	. . . . . 6 |- ( f : A -1-1-onto-> B -> ( ( z C_ B /\ z ~~ C ) -> ( ( `' f " z ) C_ A /\ ( `' f " z ) ~~ C ) ) )
64		elin 3388	. . . . . . 7 |- ( z e. ( ~P B i^i { x | x ~~ C } ) <-> ( z e. ~P B /\ z e. { x | x ~~ C } ) )
65	51	elpw 3656	. . . . . . . 8 |- ( z e. ~P B <-> z C_ B )
66		breq1 4089	. . . . . . . . 9 |- ( x = z -> ( x ~~ C <-> z ~~ C ) )
67	51, 66	elab 2948	. . . . . . . 8 |- ( z e. { x | x ~~ C } <-> z ~~ C )
68	65, 67	anbi12i 460	. . . . . . 7 |- ( ( z e. ~P B /\ z e. { x | x ~~ C } ) <-> ( z C_ B /\ z ~~ C ) )
69	64, 68	bitri 184	. . . . . 6 |- ( z e. ( ~P B i^i { x | x ~~ C } ) <-> ( z C_ B /\ z ~~ C ) )
70		elin 3388	. . . . . . 7 |- ( ( `' f " z ) e. ( ~P A i^i { x | x ~~ C } ) <-> ( ( `' f " z ) e. ~P A /\ ( `' f " z ) e. { x | x ~~ C } ) )
71	3	cnvex 5273	. . . . . . . . . 10 |- `' f e. _V
72	71	imaex 5089	. . . . . . . . 9 |- ( `' f " z ) e. _V
73	72	elpw 3656	. . . . . . . 8 |- ( ( `' f " z ) e. ~P A <-> ( `' f " z ) C_ A )
74		breq1 4089	. . . . . . . . 9 |- ( x = ( `' f " z ) -> ( x ~~ C <-> ( `' f " z ) ~~ C ) )
75	72, 74	elab 2948	. . . . . . . 8 |- ( ( `' f " z ) e. { x | x ~~ C } <-> ( `' f " z ) ~~ C )
76	73, 75	anbi12i 460	. . . . . . 7 |- ( ( ( `' f " z ) e. ~P A /\ ( `' f " z ) e. { x | x ~~ C } ) <-> ( ( `' f " z ) C_ A /\ ( `' f " z ) ~~ C ) )
77	70, 76	bitri 184	. . . . . 6 |- ( ( `' f " z ) e. ( ~P A i^i { x | x ~~ C } ) <-> ( ( `' f " z ) C_ A /\ ( `' f " z ) ~~ C ) )
78	63, 69, 77	3imtr4g 205	. . . . 5 |- ( f : A -1-1-onto-> B -> ( z e. ( ~P B i^i { x | x ~~ C } ) -> ( `' f " z ) e. ( ~P A i^i { x | x ~~ C } ) ) )
79		simpl 109	. . . . . . . . . . 11 |- ( ( z e. ~P B /\ z e. { x | x ~~ C } ) -> z e. ~P B )
80	79	elpwid 3661	. . . . . . . . . 10 |- ( ( z e. ~P B /\ z e. { x | x ~~ C } ) -> z C_ B )
81	64, 80	sylbi 121	. . . . . . . . 9 |- ( z e. ( ~P B i^i { x | x ~~ C } ) -> z C_ B )
82		imaeq2 5070	. . . . . . . . . . . 12 |- ( y = ( `' f " z ) -> ( f " y ) = ( f " ( `' f " z ) ) )
83		f1orel 5583	. . . . . . . . . . . . . . . 16 |- ( f : A -1-1-onto-> B -> Rel f )
84		dfrel2 5185	. . . . . . . . . . . . . . . 16 |- ( Rel f <-> `' `' f = f )
85	83, 84	sylib 122	. . . . . . . . . . . . . . 15 |- ( f : A -1-1-onto-> B -> `' `' f = f )
86	85	imaeq1d 5073	. . . . . . . . . . . . . 14 |- ( f : A -1-1-onto-> B -> ( `' `' f " ( `' f " z ) ) = ( f " ( `' f " z ) ) )
87	86	adantr 276	. . . . . . . . . . . . 13 |- ( ( f : A -1-1-onto-> B /\ z C_ B ) -> ( `' `' f " ( `' f " z ) ) = ( f " ( `' f " z ) ) )
88	46, 47	syl 14	. . . . . . . . . . . . . 14 |- ( f : A -1-1-onto-> B -> `' f : B -1-1-> A )
89		f1imacnv 5597	. . . . . . . . . . . . . 14 |- ( ( `' f : B -1-1-> A /\ z C_ B ) -> ( `' `' f " ( `' f " z ) ) = z )
90	88, 89	sylan 283	. . . . . . . . . . . . 13 |- ( ( f : A -1-1-onto-> B /\ z C_ B ) -> ( `' `' f " ( `' f " z ) ) = z )
91	87, 90	eqtr3d 2264	. . . . . . . . . . . 12 |- ( ( f : A -1-1-onto-> B /\ z C_ B ) -> ( f " ( `' f " z ) ) = z )
92	82, 91	sylan9eqr 2284	. . . . . . . . . . 11 |- ( ( ( f : A -1-1-onto-> B /\ z C_ B ) /\ y = ( `' f " z ) ) -> ( f " y ) = z )
93	92	eqcomd 2235	. . . . . . . . . 10 |- ( ( ( f : A -1-1-onto-> B /\ z C_ B ) /\ y = ( `' f " z ) ) -> z = ( f " y ) )
94	93	ex 115	. . . . . . . . 9 |- ( ( f : A -1-1-onto-> B /\ z C_ B ) -> ( y = ( `' f " z ) -> z = ( f " y ) ) )
95	81, 94	sylan2 286	. . . . . . . 8 |- ( ( f : A -1-1-onto-> B /\ z e. ( ~P B i^i { x | x ~~ C } ) ) -> ( y = ( `' f " z ) -> z = ( f " y ) ) )
96	95	adantrl 478	. . . . . . 7 |- ( ( f : A -1-1-onto-> B /\ ( y e. ( ~P A i^i { x | x ~~ C } ) /\ z e. ( ~P B i^i { x | x ~~ C } ) ) ) -> ( y = ( `' f " z ) -> z = ( f " y ) ) )
97		simpl 109	. . . . . . . . . . 11 |- ( ( y e. ~P A /\ y e. { x | x ~~ C } ) -> y e. ~P A )
98	97	elpwid 3661	. . . . . . . . . 10 |- ( ( y e. ~P A /\ y e. { x | x ~~ C } ) -> y C_ A )
99	32, 98	sylbi 121	. . . . . . . . 9 |- ( y e. ( ~P A i^i { x | x ~~ C } ) -> y C_ A )
100		imaeq2 5070	. . . . . . . . . . . 12 |- ( z = ( f " y ) -> ( `' f " z ) = ( `' f " ( f " y ) ) )
101		f1imacnv 5597	. . . . . . . . . . . . 13 |- ( ( f : A -1-1-> B /\ y C_ A ) -> ( `' f " ( f " y ) ) = y )
102	17, 101	sylan 283	. . . . . . . . . . . 12 |- ( ( f : A -1-1-onto-> B /\ y C_ A ) -> ( `' f " ( f " y ) ) = y )
103	100, 102	sylan9eqr 2284	. . . . . . . . . . 11 |- ( ( ( f : A -1-1-onto-> B /\ y C_ A ) /\ z = ( f " y ) ) -> ( `' f " z ) = y )
104	103	eqcomd 2235	. . . . . . . . . 10 |- ( ( ( f : A -1-1-onto-> B /\ y C_ A ) /\ z = ( f " y ) ) -> y = ( `' f " z ) )
105	104	ex 115	. . . . . . . . 9 |- ( ( f : A -1-1-onto-> B /\ y C_ A ) -> ( z = ( f " y ) -> y = ( `' f " z ) ) )
106	99, 105	sylan2 286	. . . . . . . 8 |- ( ( f : A -1-1-onto-> B /\ y e. ( ~P A i^i { x | x ~~ C } ) ) -> ( z = ( f " y ) -> y = ( `' f " z ) ) )
107	106	adantrr 479	. . . . . . 7 |- ( ( f : A -1-1-onto-> B /\ ( y e. ( ~P A i^i { x | x ~~ C } ) /\ z e. ( ~P B i^i { x | x ~~ C } ) ) ) -> ( z = ( f " y ) -> y = ( `' f " z ) ) )
108	96, 107	impbid 129	. . . . . 6 |- ( ( f : A -1-1-onto-> B /\ ( y e. ( ~P A i^i { x | x ~~ C } ) /\ z e. ( ~P B i^i { x | x ~~ C } ) ) ) -> ( y = ( `' f " z ) <-> z = ( f " y ) ) )
109	108	ex 115	. . . . 5 |- ( f : A -1-1-onto-> B -> ( ( y e. ( ~P A i^i { x | x ~~ C } ) /\ z e. ( ~P B i^i { x | x ~~ C } ) ) -> ( y = ( `' f " z ) <-> z = ( f " y ) ) ) )
110	8, 16, 45, 78, 109	en3d 6937	. . . 4 |- ( f : A -1-1-onto-> B -> ( ~P A i^i { x | x ~~ C } ) ~~ ( ~P B i^i { x | x ~~ C } ) )
111	110	exlimiv 1644	. . 3 |- ( E. f f : A -1-1-onto-> B -> ( ~P A i^i { x | x ~~ C } ) ~~ ( ~P B i^i { x | x ~~ C } ) )
112	1, 111	sylbi 121	. 2 |- ( A ~~ B -> ( ~P A i^i { x | x ~~ C } ) ~~ ( ~P B i^i { x | x ~~ C } ) )
113		df-pw 3652	. . . 4 |- ~P A = { x | x C_ A }
114	113	ineq1i 3402	. . 3 |- ( ~P A i^i { x | x ~~ C } ) = ( { x | x C_ A } i^i { x | x ~~ C } )
115		inab 3473	. . 3 |- ( { x | x C_ A } i^i { x | x ~~ C } ) = { x | ( x C_ A /\ x ~~ C ) }
116	114, 115	eqtri 2250	. 2 |- ( ~P A i^i { x | x ~~ C } ) = { x | ( x C_ A /\ x ~~ C ) }
117		df-pw 3652	. . . 4 |- ~P B = { x | x C_ B }
118	117	ineq1i 3402	. . 3 |- ( ~P B i^i { x | x ~~ C } ) = ( { x | x C_ B } i^i { x | x ~~ C } )
119		inab 3473	. . 3 |- ( { x | x C_ B } i^i { x | x ~~ C } ) = { x | ( x C_ B /\ x ~~ C ) }
120	118, 119	eqtri 2250	. 2 |- ( ~P B i^i { x | x ~~ C } ) = { x | ( x C_ B /\ x ~~ C ) }
121	112, 116, 120	3brtr3g 4119	1 |- ( A ~~ B -> { x | ( x C_ A /\ x ~~ C ) } ~~ { x | ( x C_ B /\ x ~~ C ) } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   _Vcvv 2800   i^i cin 3197   C_ wss 3198   ~Pcpw 3650   class class class wbr 4086   `'ccnv 4722   dom cdm 4723   ran crn 4724   "cima 4726   Rel wrel 4728   -1-1->wf1 5321   -onto->wfo 5322   -1-1-onto->wf1o 5323   ~~ cen 6902

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-er 6697  df-en 6905

This theorem is referenced by: (None)