Theorem hashbcval 15706

Description: Value of the "binomial set", the set of all N-element subsets of A. (Contributed by Mario Carneiro, 20-Apr-2015.)

Hypothesis

Ref	Expression
ramval.c	|- C = ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } )

Assertion

Ref	Expression
hashbcval	|- ( ( A e. V /\ N e. NN0 ) -> ( A C N ) = { x e. ~P A | ( # ` x ) = N } )

Distinct variable groups:   x, C   a, b, i, x   A, a, i, x   N, a, i, x   x, V

Allowed substitution hints:   A( b)   C( i, a, b)   N( b)   V( i, a, b)

Proof of Theorem hashbcval

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( A e. V -> A e. _V )
2		pwexg 4850	. . . . 5 |- ( A e. _V -> ~P A e. _V )
3	2	adantr 481	. . . 4 |- ( ( A e. _V /\ N e. NN0 ) -> ~P A e. _V )
4		rabexg 4812	. . . 4 |- ( ~P A e. _V -> { x e. ~P A | ( # ` x ) = N } e. _V )
5	3, 4	syl 17	. . 3 |- ( ( A e. _V /\ N e. NN0 ) -> { x e. ~P A | ( # ` x ) = N } e. _V )
6		fveq2 6191	. . . . . . 7 |- ( b = x -> ( # ` b ) = ( # ` x ) )
7	6	eqeq1d 2624	. . . . . 6 |- ( b = x -> ( ( # ` b ) = i <-> ( # ` x ) = i ) )
8	7	cbvrabv 3199	. . . . 5 |- { b e. ~P a | ( # ` b ) = i } = { x e. ~P a | ( # ` x ) = i }
9		simpl 473	. . . . . . 7 |- ( ( a = A /\ i = N ) -> a = A )
10	9	pweqd 4163	. . . . . 6 |- ( ( a = A /\ i = N ) -> ~P a = ~P A )
11		simpr 477	. . . . . . 7 |- ( ( a = A /\ i = N ) -> i = N )
12	11	eqeq2d 2632	. . . . . 6 |- ( ( a = A /\ i = N ) -> ( ( # ` x ) = i <-> ( # ` x ) = N ) )
13	10, 12	rabeqbidv 3195	. . . . 5 |- ( ( a = A /\ i = N ) -> { x e. ~P a | ( # ` x ) = i } = { x e. ~P A | ( # ` x ) = N } )
14	8, 13	syl5eq 2668	. . . 4 |- ( ( a = A /\ i = N ) -> { b e. ~P a | ( # ` b ) = i } = { x e. ~P A | ( # ` x ) = N } )
15		ramval.c	. . . 4 |- C = ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } )
16	14, 15	ovmpt2ga 6790	. . 3 |- ( ( A e. _V /\ N e. NN0 /\ { x e. ~P A | ( # ` x ) = N } e. _V ) -> ( A C N ) = { x e. ~P A | ( # ` x ) = N } )
17	5, 16	mpd3an3 1425	. 2 |- ( ( A e. _V /\ N e. NN0 ) -> ( A C N ) = { x e. ~P A | ( # ` x ) = N } )
18	1, 17	sylan 488	1 |- ( ( A e. V /\ N e. NN0 ) -> ( A C N ) = { x e. ~P A | ( # ` x ) = N } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   ~Pcpw 4158   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   NN0cn0 11292   #chash 13117

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-sep 4781  ax-nul 4789  ax-pow 4843  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  hashbccl  15707  hashbcss  15708  hashbc0  15709  hashbc2  15710  ramval  15712  ram0  15726  ramub1lem1  15730  ramub1lem2  15731