Theorem ssenen 5789

Description: Equinumerosity of equinumerous subsets of a set.

Hypotheses

Ref	Expression
ssenen.1	|- A e. _V
ssenen.2	|- B e. _V

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

Step	Hyp	Ref	Expression
1		ssenen.2	. . . 4 |- B e. _V
2	1	bren 5597	. . 3 |- (A ~~ B <-> E.f f:A-1-1-onto->B)
3		ssenen.1	. . . . . . . 8 |- A e. _V
4	3	pwex 3654	. . . . . . 7 |- ~PA e. _V
5	4	inex1 3619	. . . . . 6 |- (~PA i^i {x | x ~~ C}) e. _V
6	5	a1i 8	. . . . 5 |- (f:A-1-1-onto->B -> (~PA i^i {x | x ~~ C}) e. _V)
7		f1of1 4722	. . . . . . . . . 10 |- (f:A-1-1-onto->B -> f:A-1-1->B)
8		visset 2541	. . . . . . . . . . 11 |- y e. _V
9	8	f1imaen 5642	. . . . . . . . . 10 |- ((f:A-1-1->B /\ y C_ A) -> (f"y) ~~ y)
10	7, 9	sylan 597	. . . . . . . . 9 |- ((f:A-1-1-onto->B /\ y C_ A) -> (f"y) ~~ y)
11		entr 5634	. . . . . . . . 9 |- (((f"y) ~~ y /\ y ~~ C) -> (f"y) ~~ C)
12	10, 11	sylan 597	. . . . . . . 8 |- (((f:A-1-1-onto->B /\ y C_ A) /\ y ~~ C) -> (f"y) ~~ C)
13	12	expl 591	. . . . . . 7 |- (f:A-1-1-onto->B -> ((y C_ A /\ y ~~ C) -> (f"y) ~~ C))
14		f1ofo 4729	. . . . . . . 8 |- (f:A-1-1-onto->B -> f:A-onto->B)
15		imassrn 4396	. . . . . . . . 9 |- (f"y) C_ ran f
16		forn 4708	. . . . . . . . . 10 |- (f:A-onto->B -> ran f = B)
17	16	sseq2d 2872	. . . . . . . . 9 |- (f:A-onto->B -> ((f"y) C_ ran f <-> (f"y) C_ B))
18	15, 17	mpbii 319	. . . . . . . 8 |- (f:A-onto->B -> (f"y) C_ B)
19	14, 18	syl 13	. . . . . . 7 |- (f:A-1-1-onto->B -> (f"y) C_ B)
20	13, 19	jctild 507	. . . . . 6 |- (f:A-1-1-onto->B -> ((y C_ A /\ y ~~ C) -> ((f"y) C_ B /\ (f"y) ~~ C)))
21		elin 2999	. . . . . . 7 |- (y e. (~PA i^i {x | x ~~ C}) <-> (y e. ~PA /\ y e. {x | x ~~ C}))
22	8	elpw 3231	. . . . . . . 8 |- (y e. ~PA <-> y C_ A)
23		breq1 3510	. . . . . . . . 9 |- (x = y -> (x ~~ C <-> y ~~ C))
24	8, 23	elab 2644	. . . . . . . 8 |- (y e. {x | x ~~ C} <-> y ~~ C)
25	22, 24	anbi12i 710	. . . . . . 7 |- ((y e. ~PA /\ y e. {x | x ~~ C}) <-> (y C_ A /\ y ~~ C))
26	21, 25	bitri 279	. . . . . 6 |- (y e. (~PA i^i {x | x ~~ C}) <-> (y C_ A /\ y ~~ C))
27		elin 2999	. . . . . . 7 |- ((f"y) e. (~PB i^i {x | x ~~ C}) <-> ((f"y) e. ~PB /\ (f"y) e. {x | x ~~ C}))
28		visset 2541	. . . . . . . . . 10 |- f e. _V
29		imaexg 4397	. . . . . . . . . 10 |- (f e. _V -> (f"y) e. _V)
30	28, 29	ax-mp 7	. . . . . . . . 9 |- (f"y) e. _V
31	30	elpw 3231	. . . . . . . 8 |- ((f"y) e. ~PB <-> (f"y) C_ B)
32		breq1 3510	. . . . . . . . 9 |- (x = (f"y) -> (x ~~ C <-> (f"y) ~~ C))
33	30, 32	elab 2644	. . . . . . . 8 |- ((f"y) e. {x | x ~~ C} <-> (f"y) ~~ C)
34	31, 33	anbi12i 710	. . . . . . 7 |- (((f"y) e. ~PB /\ (f"y) e. {x | x ~~ C}) <-> ((f"y) C_ B /\ (f"y) ~~ C))
35	27, 34	bitri 279	. . . . . 6 |- ((f"y) e. (~PB i^i {x | x ~~ C}) <-> ((f"y) C_ B /\ (f"y) ~~ C))
36	20, 26, 35	3imtr4g 332	. . . . 5 |- (f:A-1-1-onto->B -> (y e. (~PA i^i {x | x ~~ C}) -> (f"y) e. (~PB i^i {x | x ~~ C})))
37		f1ocnv 4735	. . . . . . 7 |- (f:A-1-1-onto->B -> `'f:B-1-1-onto->A)
38		f1of1 4722	. . . . . . . . . . 11 |- (`'f:B-1-1-onto->A -> `'f:B-1-1->A)
39		visset 2541	. . . . . . . . . . . 12 |- z e. _V
40	39	f1imaen 5642	. . . . . . . . . . 11 |- ((`'f:B-1-1->A /\ z C_ B) -> (`'f"z) ~~ z)
41	38, 40	sylan 597	. . . . . . . . . 10 |- ((`'f:B-1-1-onto->A /\ z C_ B) -> (`'f"z) ~~ z)
42		entr 5634	. . . . . . . . . 10 |- (((`'f"z) ~~ z /\ z ~~ C) -> (`'f"z) ~~ C)
43	41, 42	sylan 597	. . . . . . . . 9 |- (((`'f:B-1-1-onto->A /\ z C_ B) /\ z ~~ C) -> (`'f"z) ~~ C)
44	43	expl 591	. . . . . . . 8 |- (`'f:B-1-1-onto->A -> ((z C_ B /\ z ~~ C) -> (`'f"z) ~~ C))
45		f1ofo 4729	. . . . . . . . 9 |- (`'f:B-1-1-onto->A -> `'f:B-onto->A)
46		imassrn 4396	. . . . . . . . . 10 |- (`'f"z) C_ ran `' f
47		forn 4708	. . . . . . . . . . 11 |- (`'f:B-onto->A -> ran `' f = A)
48	47	sseq2d 2872	. . . . . . . . . 10 |- (`'f:B-onto->A -> ((`'f"z) C_ ran `' f <-> (`'f"z) C_ A))
49	46, 48	mpbii 319	. . . . . . . . 9 |- (`'f:B-onto->A -> (`'f"z) C_ A)
50	45, 49	syl 13	. . . . . . . 8 |- (`'f:B-1-1-onto->A -> (`'f"z) C_ A)
51	44, 50	jctild 507	. . . . . . 7 |- (`'f:B-1-1-onto->A -> ((z C_ B /\ z ~~ C) -> ((`'f"z) C_ A /\ (`'f"z) ~~ C)))
52	37, 51	syl 13	. . . . . 6 |- (f:A-1-1-onto->B -> ((z C_ B /\ z ~~ C) -> ((`'f"z) C_ A /\ (`'f"z) ~~ C)))
53		elin 2999	. . . . . . 7 |- (z e. (~PB i^i {x | x ~~ C}) <-> (z e. ~PB /\ z e. {x | x ~~ C}))
54	39	elpw 3231	. . . . . . . 8 |- (z e. ~PB <-> z C_ B)
55		breq1 3510	. . . . . . . . 9 |- (x = z -> (x ~~ C <-> z ~~ C))
56	39, 55	elab 2644	. . . . . . . 8 |- (z e. {x | x ~~ C} <-> z ~~ C)
57	54, 56	anbi12i 710	. . . . . . 7 |- ((z e. ~PB /\ z e. {x | x ~~ C}) <-> (z C_ B /\ z ~~ C))
58	53, 57	bitri 279	. . . . . 6 |- (z e. (~PB i^i {x | x ~~ C}) <-> (z C_ B /\ z ~~ C))
59		elin 2999	. . . . . . 7 |- ((`'f"z) e. (~PA i^i {x | x ~~ C}) <-> ((`'f"z) e. ~PA /\ (`'f"z) e. {x | x ~~ C}))
60	28	cnvex 4528	. . . . . . . . . 10 |- `'f e. _V
61		imaexg 4397	. . . . . . . . . 10 |- (`'f e. _V -> (`'f"z) e. _V)
62	60, 61	ax-mp 7	. . . . . . . . 9 |- (`'f"z) e. _V
63	62	elpw 3231	. . . . . . . 8 |- ((`'f"z) e. ~PA <-> (`'f"z) C_ A)
64		breq1 3510	. . . . . . . . 9 |- (x = (`'f"z) -> (x ~~ C <-> (`'f"z) ~~ C))
65	62, 64	elab 2644	. . . . . . . 8 |- ((`'f"z) e. {x | x ~~ C} <-> (`'f"z) ~~ C)
66	63, 65	anbi12i 710	. . . . . . 7 |- (((`'f"z) e. ~PA /\ (`'f"z) e. {x | x ~~ C}) <-> ((`'f"z) C_ A /\ (`'f"z) ~~ C))
67	59, 66	bitri 279	. . . . . 6 |- ((`'f"z) e. (~PA i^i {x | x ~~ C}) <-> ((`'f"z) C_ A /\ (`'f"z) ~~ C))
68	52, 58, 67	3imtr4g 332	. . . . 5 |- (f:A-1-1-onto->B -> (z e. (~PB i^i {x | x ~~ C}) -> (`'f"z) e. (~PA i^i {x | x ~~ C})))
69		simpl 437	. . . . . . . . . . 11 |- ((z e. ~PB /\ z e. {x | x ~~ C}) -> z e. ~PB)
70	69, 54	sylib 242	. . . . . . . . . 10 |- ((z e. ~PB /\ z e. {x | x ~~ C}) -> z C_ B)
71	53, 70	sylbi 225	. . . . . . . . 9 |- (z e. (~PB i^i {x | x ~~ C}) -> z C_ B)
72		imaeq2 4380	. . . . . . . . . . . 12 |- (y = (`'f"z) -> (f"y) = (f"(`'f"z)))
73		f1orel 4726	. . . . . . . . . . . . . . . 16 |- (f:A-1-1-onto->B -> Rel f)
74		dfrel2 4463	. . . . . . . . . . . . . . . 16 |- (Rel f <-> `'`'f = f)
75	73, 74	sylib 242	. . . . . . . . . . . . . . 15 |- (f:A-1-1-onto->B -> `'`'f = f)
76	75	imaeq1d 4383	. . . . . . . . . . . . . 14 |- (f:A-1-1-onto->B -> (`'`'f"(`'f"z)) = (f"(`'f"z)))
77	76	adantr 447	. . . . . . . . . . . . 13 |- ((f:A-1-1-onto->B /\ z C_ B) -> (`'`'f"(`'f"z)) = (f"(`'f"z)))
78	37, 38	syl 13	. . . . . . . . . . . . . 14 |- (f:A-1-1-onto->B -> `'f:B-1-1->A)
79		f1imacnv 4739	. . . . . . . . . . . . . 14 |- ((`'f:B-1-1->A /\ z C_ B) -> (`'`'f"(`'f"z)) = z)
80	78, 79	sylan 597	. . . . . . . . . . . . 13 |- ((f:A-1-1-onto->B /\ z C_ B) -> (`'`'f"(`'f"z)) = z)
81	77, 80	eqtr3d 2175	. . . . . . . . . . . 12 |- ((f:A-1-1-onto->B /\ z C_ B) -> (f"(`'f"z)) = z)
82	72, 81	sylan9eqr 2199	. . . . . . . . . . 11 |- (((f:A-1-1-onto->B /\ z C_ B) /\ y = (`'f"z)) -> (f"y) = z)
83	82	eqcomd 2146	. . . . . . . . . 10 |- (((f:A-1-1-onto->B /\ z C_ B) /\ y = (`'f"z)) -> z = (f"y))
84	83	ex 398	. . . . . . . . 9 |- ((f:A-1-1-onto->B /\ z C_ B) -> (y = (`'f"z) -> z = (f"y)))
85	71, 84	sylan2 600	. . . . . . . 8 |- ((f:A-1-1-onto->B /\ z e. (~PB i^i {x | x ~~ C})) -> (y = (`'f"z) -> z = (f"y)))
86	85	adantrl 779	. . . . . . 7 |- ((f:A-1-1-onto->B /\ (y e. (~PA i^i {x | x ~~ C}) /\ z e. (~PB i^i {x | x ~~ C}))) -> (y = (`'f"z) -> z = (f"y)))
87		simpl 437	. . . . . . . . . . 11 |- ((y e. ~PA /\ y e. {x | x ~~ C}) -> y e. ~PA)
88	87, 22	sylib 242	. . . . . . . . . 10 |- ((y e. ~PA /\ y e. {x | x ~~ C}) -> y C_ A)
89	21, 88	sylbi 225	. . . . . . . . 9 |- (y e. (~PA i^i {x | x ~~ C}) -> y C_ A)
90		imaeq2 4380	. . . . . . . . . . . 12 |- (z = (f"y) -> (`'f"z) = (`'f"(f"y)))
91		f1imacnv 4739	. . . . . . . . . . . . 13 |- ((f:A-1-1->B /\ y C_ A) -> (`'f"(f"y)) = y)
92	7, 91	sylan 597	. . . . . . . . . . . 12 |- ((f:A-1-1-onto->B /\ y C_ A) -> (`'f"(f"y)) = y)
93	90, 92	sylan9eqr 2199	. . . . . . . . . . 11 |- (((f:A-1-1-onto->B /\ y C_ A) /\ z = (f"y)) -> (`'f"z) = y)
94	93	eqcomd 2146	. . . . . . . . . 10 |- (((f:A-1-1-onto->B /\ y C_ A) /\ z = (f"y)) -> y = (`'f"z))
95	94	ex 398	. . . . . . . . 9 |- ((f:A-1-1-onto->B /\ y C_ A) -> (z = (f"y) -> y = (`'f"z)))
96	89, 95	sylan2 600	. . . . . . . 8 |- ((f:A-1-1-onto->B /\ y e. (~PA i^i {x | x ~~ C})) -> (z = (f"y) -> y = (`'f"z)))
97	96	adantrr 781	. . . . . . 7 |- ((f:A-1-1-onto->B /\ (y e. (~PA i^i {x | x ~~ C}) /\ z e. (~PB i^i {x | x ~~ C}))) -> (z = (f"y) -> y = (`'f"z)))
98	86, 97	impbid 235	. . . . . 6 |- ((f:A-1-1-onto->B /\ (y e. (~PA i^i {x | x ~~ C}) /\ z e. (~PB i^i {x | x ~~ C}))) -> (y = (`'f"z) <-> z = (f"y)))
99	98	ex 398	. . . . 5 |- (f:A-1-1-onto->B -> ((y e. (~PA i^i {x | x ~~ C}) /\ z e. (~PB i^i {x | x ~~ C})) -> (y = (`'f"z) <-> z = (f"y))))
100	6, 36, 68, 99	en3d 5621	. . . 4 |- (f:A-1-1-onto->B -> (~PA i^i {x | x ~~ C}) ~~ (~PB i^i {x | x ~~ C}))
101	100	19.23aiv 1943	. . 3 |- (E.f f:A-1-1-onto->B -> (~PA i^i {x | x ~~ C}) ~~ (~PB i^i {x | x ~~ C}))
102	2, 101	sylbi 225	. 2 |- (A ~~ B -> (~PA i^i {x | x ~~ C}) ~~ (~PB i^i {x | x ~~ C}))
103		df-pw 3229	. . . 4 |- ~PA = {x | x C_ A}
104	103	ineq1i 3005	. . 3 |- (~PA i^i {x | x ~~ C}) = ({x | x C_ A} i^i {x | x ~~ C})
105		inab 3070	. . 3 |- ({x | x C_ A} i^i {x | x ~~ C}) = {x | (x C_ A /\ x ~~ C)}
106	104, 105	eqtri 2161	. 2 |- (~PA i^i {x | x ~~ C}) = {x | (x C_ A /\ x ~~ C)}
107		df-pw 3229	. . . 4 |- ~PB = {x | x C_ B}
108	107	ineq1i 3005	. . 3 |- (~PB i^i {x | x ~~ C}) = ({x | x C_ B} i^i {x | x ~~ C})
109		inab 3070	. . 3 |- ({x | x C_ B} i^i {x | x ~~ C}) = {x | (x C_ B /\ x ~~ C)}
110	108, 109	eqtri 2161	. 2 |- (~PB i^i {x | x ~~ C}) = {x | (x C_ B /\ x ~~ C)}
111	102, 106, 110	3brtr3g 3538	1 |- (A ~~ B -> {x | (x C_ A /\ x ~~ C)} ~~ {x | (x C_ B /\ x ~~ C)})

Syntax hints:   -> wi 3   <-> wb 219   /\ wa 337   = wceq 1586   e. wcel 1588  E.wex 1615  {cab 2128  _Vcvv 2538   i^i cin 2826   C_ wss 2827  ~Pcpw 3227   class class class wbr 3507  `'ccnv 4118  ran crn 4120  "cima 4122  Rel wrel 4124  -1-1->wf1 4128  -onto->wfo 4129  -1-1-onto->wf1o 4130   ~~ cen 5584

This theorem is referenced by:  infmap2 9244

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1592  ax-gen 1593  ax-8 1594  ax-9 1595  ax-10 1596  ax-11 1597  ax-12 1598  ax-13 1599  ax-14 1600  ax-17 1605  ax-4 1608  ax-5o 1610  ax-6o 1613  ax-9o 1763  ax-10o 1781  ax-16 1854  ax-11o 1864  ax-ext 2123  ax-rep 3596  ax-sep 3606  ax-nul 3613  ax-pow 3649  ax-pr 3687  ax-un 3929

This theorem depends on definitions:  df-bi 220  df-or 338  df-an 339  df-3an 1104  df-ex 1616  df-sb 1816  df-eu 2041  df-mo 2042  df-clab 2129  df-cleq 2134  df-clel 2137  df-ne 2268  df-ral 2359  df-rex 2360  df-v 2540  df-dif 2830  df-un 2832  df-in 2834  df-ss 2836  df-nul 3083  df-pw 3229  df-sn 3242  df-pr 3243  df-op 3246  df-uni 3367  df-br 3508  df-opab 3566  df-id 3747  df-xp 4133  df-rel 4134  df-cnv 4135  df-co 4136  df-dm 4137  df-rn 4138  df-res 4139  df-ima 4140  df-fun 4141  df-fn 4142  df-f 4143  df-f1 4144  df-fo 4145  df-f1o 4146  df-er 5479  df-en 5588

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