Theorem hashfibc 11200

Description: The binomial coefficient counts the number of subsets of a finite set of a given size. This is Metamath 100 proof #58 (formula for the number of combinations). For more on the notation for subsets of a given size, see sseqn 11196. (Contributed by Mario Carneiro, 13-Jul-2014.)

Assertion

Ref	Expression
hashfibc	|- ( ( A e. Fin /\ K e. ZZ ) -> ( (♯ ` A ) _C K ) = (♯ ` { x e. ( ~P A i^i Fin ) | (♯ ` x ) = K } ) )

Distinct variable groups:   x, A   x, K

Proof of Theorem hashfibc

Dummy variables k w y z j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5669	. . . . . 6 |- ( w = (/) -> (♯ ` w ) = (♯ ` (/) ) )
2	1	oveq1d 6064	. . . . 5 |- ( w = (/) -> ( (♯ ` w ) _C k ) = ( (♯ ` (/) ) _C k ) )
3		pweq 3671	. . . . . . . 8 |- ( w = (/) -> ~P w = ~P (/) )
4	3	ineq1d 3420	. . . . . . 7 |- ( w = (/) -> ( ~P w i^i Fin ) = ( ~P (/) i^i Fin ) )
5	4	rabeqdv 2806	. . . . . 6 |- ( w = (/) -> { x e. ( ~P w i^i Fin ) | (♯ ` x ) = k } = { x e. ( ~P (/) i^i Fin ) | (♯ ` x ) = k } )
6	5	fveq2d 5673	. . . . 5 |- ( w = (/) -> (♯ ` { x e. ( ~P w i^i Fin ) | (♯ ` x ) = k } ) = (♯ ` { x e. ( ~P (/) i^i Fin ) | (♯ ` x ) = k } ) )
7	2, 6	eqeq12d 2247	. . . 4 |- ( w = (/) -> ( ( (♯ ` w ) _C k ) = (♯ ` { x e. ( ~P w i^i Fin ) | (♯ ` x ) = k } ) <-> ( (♯ ` (/) ) _C k ) = (♯ ` { x e. ( ~P (/) i^i Fin ) | (♯ ` x ) = k } ) ) )
8	7	ralbidv 2542	. . 3 |- ( w = (/) -> ( A. k e. ZZ ( (♯ ` w ) _C k ) = (♯ ` { x e. ( ~P w i^i Fin ) | (♯ ` x ) = k } ) <-> A. k e. ZZ ( (♯ ` (/) ) _C k ) = (♯ ` { x e. ( ~P (/) i^i Fin ) | (♯ ` x ) = k } ) ) )
9		fveq2 5669	. . . . . 6 |- ( w = y -> (♯ ` w ) = (♯ ` y ) )
10	9	oveq1d 6064	. . . . 5 |- ( w = y -> ( (♯ ` w ) _C k ) = ( (♯ ` y ) _C k ) )
11		pweq 3671	. . . . . . . 8 |- ( w = y -> ~P w = ~P y )
12	11	ineq1d 3420	. . . . . . 7 |- ( w = y -> ( ~P w i^i Fin ) = ( ~P y i^i Fin ) )
13	12	rabeqdv 2806	. . . . . 6 |- ( w = y -> { x e. ( ~P w i^i Fin ) | (♯ ` x ) = k } = { x e. ( ~P y i^i Fin ) | (♯ ` x ) = k } )
14	13	fveq2d 5673	. . . . 5 |- ( w = y -> (♯ ` { x e. ( ~P w i^i Fin ) | (♯ ` x ) = k } ) = (♯ ` { x e. ( ~P y i^i Fin ) | (♯ ` x ) = k } ) )
15	10, 14	eqeq12d 2247	. . . 4 |- ( w = y -> ( ( (♯ ` w ) _C k ) = (♯ ` { x e. ( ~P w i^i Fin ) | (♯ ` x ) = k } ) <-> ( (♯ ` y ) _C k ) = (♯ ` { x e. ( ~P y i^i Fin ) | (♯ ` x ) = k } ) ) )
16	15	ralbidv 2542	. . 3 |- ( w = y -> ( A. k e. ZZ ( (♯ ` w ) _C k ) = (♯ ` { x e. ( ~P w i^i Fin ) | (♯ ` x ) = k } ) <-> A. k e. ZZ ( (♯ ` y ) _C k ) = (♯ ` { x e. ( ~P y i^i Fin ) | (♯ ` x ) = k } ) ) )
17		fveq2 5669	. . . . . 6 |- ( w = ( y u. { z } ) -> (♯ ` w ) = (♯ ` ( y u. { z } ) ) )
18	17	oveq1d 6064	. . . . 5 |- ( w = ( y u. { z } ) -> ( (♯ ` w ) _C k ) = ( (♯ ` ( y u. { z } ) ) _C k ) )
19		pweq 3671	. . . . . . . 8 |- ( w = ( y u. { z } ) -> ~P w = ~P ( y u. { z } ) )
20	19	ineq1d 3420	. . . . . . 7 |- ( w = ( y u. { z } ) -> ( ~P w i^i Fin ) = ( ~P ( y u. { z } ) i^i Fin ) )
21	20	rabeqdv 2806	. . . . . 6 |- ( w = ( y u. { z } ) -> { x e. ( ~P w i^i Fin ) | (♯ ` x ) = k } = { x e. ( ~P ( y u. { z } ) i^i Fin ) | (♯ ` x ) = k } )
22	21	fveq2d 5673	. . . . 5 |- ( w = ( y u. { z } ) -> (♯ ` { x e. ( ~P w i^i Fin ) | (♯ ` x ) = k } ) = (♯ ` { x e. ( ~P ( y u. { z } ) i^i Fin ) | (♯ ` x ) = k } ) )
23	18, 22	eqeq12d 2247	. . . 4 |- ( w = ( y u. { z } ) -> ( ( (♯ ` w ) _C k ) = (♯ ` { x e. ( ~P w i^i Fin ) | (♯ ` x ) = k } ) <-> ( (♯ ` ( y u. { z } ) ) _C k ) = (♯ ` { x e. ( ~P ( y u. { z } ) i^i Fin ) | (♯ ` x ) = k } ) ) )
24	23	ralbidv 2542	. . 3 |- ( w = ( y u. { z } ) -> ( A. k e. ZZ ( (♯ ` w ) _C k ) = (♯ ` { x e. ( ~P w i^i Fin ) | (♯ ` x ) = k } ) <-> A. k e. ZZ ( (♯ ` ( y u. { z } ) ) _C k ) = (♯ ` { x e. ( ~P ( y u. { z } ) i^i Fin ) | (♯ ` x ) = k } ) ) )
25		fveq2 5669	. . . . . 6 |- ( w = A -> (♯ ` w ) = (♯ ` A ) )
26	25	oveq1d 6064	. . . . 5 |- ( w = A -> ( (♯ ` w ) _C k ) = ( (♯ ` A ) _C k ) )
27		pweq 3671	. . . . . . . 8 |- ( w = A -> ~P w = ~P A )
28	27	ineq1d 3420	. . . . . . 7 |- ( w = A -> ( ~P w i^i Fin ) = ( ~P A i^i Fin ) )
29	28	rabeqdv 2806	. . . . . 6 |- ( w = A -> { x e. ( ~P w i^i Fin ) | (♯ ` x ) = k } = { x e. ( ~P A i^i Fin ) | (♯ ` x ) = k } )
30	29	fveq2d 5673	. . . . 5 |- ( w = A -> (♯ ` { x e. ( ~P w i^i Fin ) | (♯ ` x ) = k } ) = (♯ ` { x e. ( ~P A i^i Fin ) | (♯ ` x ) = k } ) )
31	26, 30	eqeq12d 2247	. . . 4 |- ( w = A -> ( ( (♯ ` w ) _C k ) = (♯ ` { x e. ( ~P w i^i Fin ) | (♯ ` x ) = k } ) <-> ( (♯ ` A ) _C k ) = (♯ ` { x e. ( ~P A i^i Fin ) | (♯ ` x ) = k } ) ) )
32	31	ralbidv 2542	. . 3 |- ( w = A -> ( A. k e. ZZ ( (♯ ` w ) _C k ) = (♯ ` { x e. ( ~P w i^i Fin ) | (♯ ` x ) = k } ) <-> A. k e. ZZ ( (♯ ` A ) _C k ) = (♯ ` { x e. ( ~P A i^i Fin ) | (♯ ` x ) = k } ) ) )
33		0nn0 9507	. . . . . . . . 9 |- 0 e. NN0
34		bcn0 11113	. . . . . . . . 9 |- ( 0 e. NN0 -> ( 0 _C 0 ) = 1 )
35	33, 34	ax-mp 5	. . . . . . . 8 |- ( 0 _C 0 ) = 1
36		hash0 11154	. . . . . . . . . 10 |- (♯ ` (/) ) = 0
37	36	a1i 9	. . . . . . . . 9 |- ( k e. ( 0 ... 0 ) -> (♯ ` (/) ) = 0 )
38		elfz1eq 10365	. . . . . . . . 9 |- ( k e. ( 0 ... 0 ) -> k = 0 )
39	37, 38	oveq12d 6067	. . . . . . . 8 |- ( k e. ( 0 ... 0 ) -> ( (♯ ` (/) ) _C k ) = ( 0 _C 0 ) )
40	38	eqcomd 2238	. . . . . . . . . . . . 13 |- ( k e. ( 0 ... 0 ) -> 0 = k )
41		pw0 3840	. . . . . . . . . . . . . . 15 |- ~P (/) = { (/) }
42	41	raleqi 2744	. . . . . . . . . . . . . 14 |- ( A. x e. ~P (/) (♯ ` x ) = k <-> A. x e. { (/) } (♯ ` x ) = k )
43		0ex 4236	. . . . . . . . . . . . . . 15 |- (/) e. _V
44		fveq2 5669	. . . . . . . . . . . . . . . . 17 |- ( x = (/) -> (♯ ` x ) = (♯ ` (/) ) )
45	44, 36	eqtrdi 2281	. . . . . . . . . . . . . . . 16 |- ( x = (/) -> (♯ ` x ) = 0 )
46	45	eqeq1d 2241	. . . . . . . . . . . . . . 15 |- ( x = (/) -> ( (♯ ` x ) = k <-> 0 = k ) )
47	43, 46	ralsn 3731	. . . . . . . . . . . . . 14 |- ( A. x e. { (/) } (♯ ` x ) = k <-> 0 = k )
48	42, 47	bitri 184	. . . . . . . . . . . . 13 |- ( A. x e. ~P (/) (♯ ` x ) = k <-> 0 = k )
49	40, 48	sylibr 134	. . . . . . . . . . . 12 |- ( k e. ( 0 ... 0 ) -> A. x e. ~P (/) (♯ ` x ) = k )
50		rabid2 2720	. . . . . . . . . . . 12 |- ( ~P (/) = { x e. ~P (/) | (♯ ` x ) = k } <-> A. x e. ~P (/) (♯ ` x ) = k )
51	49, 50	sylibr 134	. . . . . . . . . . 11 |- ( k e. ( 0 ... 0 ) -> ~P (/) = { x e. ~P (/) | (♯ ` x ) = k } )
52	51, 41	eqtr3di 2280	. . . . . . . . . 10 |- ( k e. ( 0 ... 0 ) -> { x e. ~P (/) | (♯ ` x ) = k } = { (/) } )
53	52	fveq2d 5673	. . . . . . . . 9 |- ( k e. ( 0 ... 0 ) -> (♯ ` { x e. ~P (/) | (♯ ` x ) = k } ) = (♯ ` { (/) } ) )
54		hashsng 11156	. . . . . . . . . 10 |- ( (/) e. _V -> (♯ ` { (/) } ) = 1 )
55	43, 54	ax-mp 5	. . . . . . . . 9 |- (♯ ` { (/) } ) = 1
56	53, 55	eqtrdi 2281	. . . . . . . 8 |- ( k e. ( 0 ... 0 ) -> (♯ ` { x e. ~P (/) | (♯ ` x ) = k } ) = 1 )
57	35, 39, 56	3eqtr4a 2291	. . . . . . 7 |- ( k e. ( 0 ... 0 ) -> ( (♯ ` (/) ) _C k ) = (♯ ` { x e. ~P (/) | (♯ ` x ) = k } ) )
58	57	adantl 277	. . . . . 6 |- ( ( k e. ZZ /\ k e. ( 0 ... 0 ) ) -> ( (♯ ` (/) ) _C k ) = (♯ ` { x e. ~P (/) | (♯ ` x ) = k } ) )
59	36	oveq1i 6059	. . . . . . 7 |- ( (♯ ` (/) ) _C k ) = ( 0 _C k )
60		bcval3 11109	. . . . . . . . 9 |- ( ( 0 e. NN0 /\ k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> ( 0 _C k ) = 0 )
61	33, 60	mp3an1 1361	. . . . . . . 8 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> ( 0 _C k ) = 0 )
62		id 19	. . . . . . . . . . . . . . 15 |- ( 0 = k -> 0 = k )
63		0z 9584	. . . . . . . . . . . . . . . 16 |- 0 e. ZZ
64		elfz3 10364	. . . . . . . . . . . . . . . 16 |- ( 0 e. ZZ -> 0 e. ( 0 ... 0 ) )
65	63, 64	ax-mp 5	. . . . . . . . . . . . . . 15 |- 0 e. ( 0 ... 0 )
66	62, 65	eqeltrrdi 2324	. . . . . . . . . . . . . 14 |- ( 0 = k -> k e. ( 0 ... 0 ) )
67	66	con3i 637	. . . . . . . . . . . . 13 |- ( -. k e. ( 0 ... 0 ) -> -. 0 = k )
68	67	adantl 277	. . . . . . . . . . . 12 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> -. 0 = k )
69	41	raleqi 2744	. . . . . . . . . . . . 13 |- ( A. x e. ~P (/) -. (♯ ` x ) = k <-> A. x e. { (/) } -. (♯ ` x ) = k )
70	46	notbid 673	. . . . . . . . . . . . . 14 |- ( x = (/) -> ( -. (♯ ` x ) = k <-> -. 0 = k ) )
71	43, 70	ralsn 3731	. . . . . . . . . . . . 13 |- ( A. x e. { (/) } -. (♯ ` x ) = k <-> -. 0 = k )
72	69, 71	bitri 184	. . . . . . . . . . . 12 |- ( A. x e. ~P (/) -. (♯ ` x ) = k <-> -. 0 = k )
73	68, 72	sylibr 134	. . . . . . . . . . 11 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> A. x e. ~P (/) -. (♯ ` x ) = k )
74		rabeq0 3537	. . . . . . . . . . 11 |- ( { x e. ~P (/) | (♯ ` x ) = k } = (/) <-> A. x e. ~P (/) -. (♯ ` x ) = k )
75	73, 74	sylibr 134	. . . . . . . . . 10 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> { x e. ~P (/) | (♯ ` x ) = k } = (/) )
76	75	fveq2d 5673	. . . . . . . . 9 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> (♯ ` { x e. ~P (/) | (♯ ` x ) = k } ) = (♯ ` (/) ) )
77	76, 36	eqtrdi 2281	. . . . . . . 8 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> (♯ ` { x e. ~P (/) | (♯ ` x ) = k } ) = 0 )
78	61, 77	eqtr4d 2268	. . . . . . 7 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> ( 0 _C k ) = (♯ ` { x e. ~P (/) | (♯ ` x ) = k } ) )
79	59, 78	eqtrid 2277	. . . . . 6 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> ( (♯ ` (/) ) _C k ) = (♯ ` { x e. ~P (/) | (♯ ` x ) = k } ) )
80		0zd 9585	. . . . . . . 8 |- ( k e. ZZ -> 0 e. ZZ )
81		fzdcel 10370	. . . . . . . 8 |- ( ( k e. ZZ /\ 0 e. ZZ /\ 0 e. ZZ ) -> DECID k e. ( 0 ... 0 ) )
82	80, 80, 81	mpd3an23 1376	. . . . . . 7 |- ( k e. ZZ -> DECID k e. ( 0 ... 0 ) )
83		exmiddc 844	. . . . . . 7 |- (DECID k e. ( 0 ... 0 ) -> ( k e. ( 0 ... 0 ) \/ -. k e. ( 0 ... 0 ) ) )
84	82, 83	syl 14	. . . . . 6 |- ( k e. ZZ -> ( k e. ( 0 ... 0 ) \/ -. k e. ( 0 ... 0 ) ) )
85	58, 79, 84	mpjaodan 806	. . . . 5 |- ( k e. ZZ -> ( (♯ ` (/) ) _C k ) = (♯ ` { x e. ~P (/) | (♯ ` x ) = k } ) )
86		velsn 3705	. . . . . . . . . . 11 |- ( x e. { (/) } <-> x = (/) )
87		0fi 7140	. . . . . . . . . . . 12 |- (/) e. Fin
88		eleq1 2295	. . . . . . . . . . . 12 |- ( x = (/) -> ( x e. Fin <-> (/) e. Fin ) )
89	87, 88	mpbiri 168	. . . . . . . . . . 11 |- ( x = (/) -> x e. Fin )
90	86, 89	sylbi 121	. . . . . . . . . 10 |- ( x e. { (/) } -> x e. Fin )
91	90, 41	eleq2s 2327	. . . . . . . . 9 |- ( x e. ~P (/) -> x e. Fin )
92	91	ssriv 3241	. . . . . . . 8 |- ~P (/) C_ Fin
93		dfss 3224	. . . . . . . 8 |- ( ~P (/) C_ Fin <-> ~P (/) = ( ~P (/) i^i Fin ) )
94	92, 93	mpbi 145	. . . . . . 7 |- ~P (/) = ( ~P (/) i^i Fin )
95	94	rabeqi 2805	. . . . . 6 |- { x e. ~P (/) | (♯ ` x ) = k } = { x e. ( ~P (/) i^i Fin ) | (♯ ` x ) = k }
96	95	fveq2i 5672	. . . . 5 |- (♯ ` { x e. ~P (/) | (♯ ` x ) = k } ) = (♯ ` { x e. ( ~P (/) i^i Fin ) | (♯ ` x ) = k } )
97	85, 96	eqtrdi 2281	. . . 4 |- ( k e. ZZ -> ( (♯ ` (/) ) _C k ) = (♯ ` { x e. ( ~P (/) i^i Fin ) | (♯ ` x ) = k } ) )
98	97	rgen 2595	. . 3 |- A. k e. ZZ ( (♯ ` (/) ) _C k ) = (♯ ` { x e. ( ~P (/) i^i Fin ) | (♯ ` x ) = k } )
99		oveq2 6057	. . . . . 6 |- ( k = j -> ( (♯ ` y ) _C k ) = ( (♯ ` y ) _C j ) )
100		eqeq2 2242	. . . . . . . . 9 |- ( k = j -> ( (♯ ` x ) = k <-> (♯ ` x ) = j ) )
101	100	rabbidv 2801	. . . . . . . 8 |- ( k = j -> { x e. ( ~P y i^i Fin ) | (♯ ` x ) = k } = { x e. ( ~P y i^i Fin ) | (♯ ` x ) = j } )
102		fveqeq2 5678	. . . . . . . . 9 |- ( x = z -> ( (♯ ` x ) = j <-> (♯ ` z ) = j ) )
103	102	cbvrabv 2811	. . . . . . . 8 |- { x e. ( ~P y i^i Fin ) | (♯ ` x ) = j } = { z e. ( ~P y i^i Fin ) | (♯ ` z ) = j }
104	101, 103	eqtrdi 2281	. . . . . . 7 |- ( k = j -> { x e. ( ~P y i^i Fin ) | (♯ ` x ) = k } = { z e. ( ~P y i^i Fin ) | (♯ ` z ) = j } )
105	104	fveq2d 5673	. . . . . 6 |- ( k = j -> (♯ ` { x e. ( ~P y i^i Fin ) | (♯ ` x ) = k } ) = (♯ ` { z e. ( ~P y i^i Fin ) | (♯ ` z ) = j } ) )
106	99, 105	eqeq12d 2247	. . . . 5 |- ( k = j -> ( ( (♯ ` y ) _C k ) = (♯ ` { x e. ( ~P y i^i Fin ) | (♯ ` x ) = k } ) <-> ( (♯ ` y ) _C j ) = (♯ ` { z e. ( ~P y i^i Fin ) | (♯ ` z ) = j } ) ) )
107	106	cbvralvw 2781	. . . 4 |- ( A. k e. ZZ ( (♯ ` y ) _C k ) = (♯ ` { x e. ( ~P y i^i Fin ) | (♯ ` x ) = k } ) <-> A. j e. ZZ ( (♯ ` y ) _C j ) = (♯ ` { z e. ( ~P y i^i Fin ) | (♯ ` z ) = j } ) )
108		simpll 527	. . . . . . 7 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( k e. ZZ /\ A. j e. ZZ ( (♯ ` y ) _C j ) = (♯ ` { z e. ( ~P y i^i Fin ) | (♯ ` z ) = j } ) ) ) -> y e. Fin )
109		simplr 529	. . . . . . 7 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( k e. ZZ /\ A. j e. ZZ ( (♯ ` y ) _C j ) = (♯ ` { z e. ( ~P y i^i Fin ) | (♯ ` z ) = j } ) ) ) -> -. z e. y )
110		simprr 533	. . . . . . . 8 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( k e. ZZ /\ A. j e. ZZ ( (♯ ` y ) _C j ) = (♯ ` { z e. ( ~P y i^i Fin ) | (♯ ` z ) = j } ) ) ) -> A. j e. ZZ ( (♯ ` y ) _C j ) = (♯ ` { z e. ( ~P y i^i Fin ) | (♯ ` z ) = j } ) )
111	103	fveq2i 5672	. . . . . . . . . 10 |- (♯ ` { x e. ( ~P y i^i Fin ) | (♯ ` x ) = j } ) = (♯ ` { z e. ( ~P y i^i Fin ) | (♯ ` z ) = j } )
112	111	eqeq2i 2243	. . . . . . . . 9 |- ( ( (♯ ` y ) _C j ) = (♯ ` { x e. ( ~P y i^i Fin ) | (♯ ` x ) = j } ) <-> ( (♯ ` y ) _C j ) = (♯ ` { z e. ( ~P y i^i Fin ) | (♯ ` z ) = j } ) )
113	112	ralbii 2548	. . . . . . . 8 |- ( A. j e. ZZ ( (♯ ` y ) _C j ) = (♯ ` { x e. ( ~P y i^i Fin ) | (♯ ` x ) = j } ) <-> A. j e. ZZ ( (♯ ` y ) _C j ) = (♯ ` { z e. ( ~P y i^i Fin ) | (♯ ` z ) = j } ) )
114	110, 113	sylibr 134	. . . . . . 7 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( k e. ZZ /\ A. j e. ZZ ( (♯ ` y ) _C j ) = (♯ ` { z e. ( ~P y i^i Fin ) | (♯ ` z ) = j } ) ) ) -> A. j e. ZZ ( (♯ ` y ) _C j ) = (♯ ` { x e. ( ~P y i^i Fin ) | (♯ ` x ) = j } ) )
115		simprl 531	. . . . . . 7 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( k e. ZZ /\ A. j e. ZZ ( (♯ ` y ) _C j ) = (♯ ` { z e. ( ~P y i^i Fin ) | (♯ ` z ) = j } ) ) ) -> k e. ZZ )
116	108, 109, 114, 115	hashfibclem 11199	. . . . . 6 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( k e. ZZ /\ A. j e. ZZ ( (♯ ` y ) _C j ) = (♯ ` { z e. ( ~P y i^i Fin ) | (♯ ` z ) = j } ) ) ) -> ( (♯ ` ( y u. { z } ) ) _C k ) = (♯ ` { x e. ( ~P ( y u. { z } ) i^i Fin ) | (♯ ` x ) = k } ) )
117	116	expr 375	. . . . 5 |- ( ( ( y e. Fin /\ -. z e. y ) /\ k e. ZZ ) -> ( A. j e. ZZ ( (♯ ` y ) _C j ) = (♯ ` { z e. ( ~P y i^i Fin ) | (♯ ` z ) = j } ) -> ( (♯ ` ( y u. { z } ) ) _C k ) = (♯ ` { x e. ( ~P ( y u. { z } ) i^i Fin ) | (♯ ` x ) = k } ) ) )
118	117	ralrimdva 2622	. . . 4 |- ( ( y e. Fin /\ -. z e. y ) -> ( A. j e. ZZ ( (♯ ` y ) _C j ) = (♯ ` { z e. ( ~P y i^i Fin ) | (♯ ` z ) = j } ) -> A. k e. ZZ ( (♯ ` ( y u. { z } ) ) _C k ) = (♯ ` { x e. ( ~P ( y u. { z } ) i^i Fin ) | (♯ ` x ) = k } ) ) )
119	107, 118	biimtrid 152	. . 3 |- ( ( y e. Fin /\ -. z e. y ) -> ( A. k e. ZZ ( (♯ ` y ) _C k ) = (♯ ` { x e. ( ~P y i^i Fin ) | (♯ ` x ) = k } ) -> A. k e. ZZ ( (♯ ` ( y u. { z } ) ) _C k ) = (♯ ` { x e. ( ~P ( y u. { z } ) i^i Fin ) | (♯ ` x ) = k } ) ) )
120	8, 16, 24, 32, 98, 119	findcard2s 7146	. 2 |- ( A e. Fin -> A. k e. ZZ ( (♯ ` A ) _C k ) = (♯ ` { x e. ( ~P A i^i Fin ) | (♯ ` x ) = k } ) )
121		oveq2 6057	. . . 4 |- ( k = K -> ( (♯ ` A ) _C k ) = ( (♯ ` A ) _C K ) )
122		eqeq2 2242	. . . . . 6 |- ( k = K -> ( (♯ ` x ) = k <-> (♯ ` x ) = K ) )
123	122	rabbidv 2801	. . . . 5 |- ( k = K -> { x e. ( ~P A i^i Fin ) | (♯ ` x ) = k } = { x e. ( ~P A i^i Fin ) | (♯ ` x ) = K } )
124	123	fveq2d 5673	. . . 4 |- ( k = K -> (♯ ` { x e. ( ~P A i^i Fin ) | (♯ ` x ) = k } ) = (♯ ` { x e. ( ~P A i^i Fin ) | (♯ ` x ) = K } ) )
125	121, 124	eqeq12d 2247	. . 3 |- ( k = K -> ( ( (♯ ` A ) _C k ) = (♯ ` { x e. ( ~P A i^i Fin ) | (♯ ` x ) = k } ) <-> ( (♯ ` A ) _C K ) = (♯ ` { x e. ( ~P A i^i Fin ) | (♯ ` x ) = K } ) ) )
126	125	rspccva 2919	. 2 |- ( ( A. k e. ZZ ( (♯ ` A ) _C k ) = (♯ ` { x e. ( ~P A i^i Fin ) | (♯ ` x ) = k } ) /\ K e. ZZ ) -> ( (♯ ` A ) _C K ) = (♯ ` { x e. ( ~P A i^i Fin ) | (♯ ` x ) = K } ) )
127	120, 126	sylan 283	1 |- ( ( A e. Fin /\ K e. ZZ ) -> ( (♯ ` A ) _C K ) = (♯ ` { x e. ( ~P A i^i Fin ) | (♯ ` x ) = K } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716  DECID wdc 842   = wceq 1398   e. wcel 2203   A.wral 2520   {crab 2524   _Vcvv 2812   u. cun 3208   i^i cin 3209   C_ wss 3210   (/)c0 3507   ~Pcpw 3668   {csn 3688   ` cfv 5351  (class class class)co 6049   Fincfn 6974   0cc0 8123   1c1 8124   NN0cn0 9492   ZZcz 9573   ...cfz 10338   _C cbc 11105  ♯chash 11133

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240  ax-pre-mulext 8241

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-frec 6621  df-1o 6646  df-2o 6647  df-oadd 6650  df-er 6766  df-map 6883  df-en 6975  df-dom 6976  df-fin 6977  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-reap 8845  df-ap 8852  df-div 8943  df-inn 9234  df-n0 9493  df-z 9574  df-uz 9850  df-q 9948  df-rp 9983  df-fz 10339  df-seqfrec 10806  df-fac 11084  df-bc 11106  df-ihash 11134

This theorem is referenced by: (None)