Theorem hashbcss 15708

Description: Subset relation for the binomial set. (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
hashbcss	|- ( ( A e. V /\ B C_ A /\ N e. NN0 ) -> ( B C N ) C_ ( A C N ) )

Distinct variable groups:   a, b, i   A, a, i   B, a, i   N, a, i

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

Proof of Theorem hashbcss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1062	. . . 4 |- ( ( A e. V /\ B C_ A /\ N e. NN0 ) -> B C_ A )
2		sspwb 4917	. . . 4 |- ( B C_ A <-> ~P B C_ ~P A )
3	1, 2	sylib 208	. . 3 |- ( ( A e. V /\ B C_ A /\ N e. NN0 ) -> ~P B C_ ~P A )
4		rabss2 3685	. . 3 |- ( ~P B C_ ~P A -> { x e. ~P B | ( # ` x ) = N } C_ { x e. ~P A | ( # ` x ) = N } )
5	3, 4	syl 17	. 2 |- ( ( A e. V /\ B C_ A /\ N e. NN0 ) -> { x e. ~P B | ( # ` x ) = N } C_ { x e. ~P A | ( # ` x ) = N } )
6		simp1 1061	. . . 4 |- ( ( A e. V /\ B C_ A /\ N e. NN0 ) -> A e. V )
7	6, 1	ssexd 4805	. . 3 |- ( ( A e. V /\ B C_ A /\ N e. NN0 ) -> B e. _V )
8		simp3 1063	. . 3 |- ( ( A e. V /\ B C_ A /\ N e. NN0 ) -> N e. NN0 )
9		ramval.c	. . . 4 |- C = ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } )
10	9	hashbcval 15706	. . 3 |- ( ( B e. _V /\ N e. NN0 ) -> ( B C N ) = { x e. ~P B | ( # ` x ) = N } )
11	7, 8, 10	syl2anc 693	. 2 |- ( ( A e. V /\ B C_ A /\ N e. NN0 ) -> ( B C N ) = { x e. ~P B | ( # ` x ) = N } )
12	9	hashbcval 15706	. . 3 |- ( ( A e. V /\ N e. NN0 ) -> ( A C N ) = { x e. ~P A | ( # ` x ) = N } )
13	12	3adant2 1080	. 2 |- ( ( A e. V /\ B C_ A /\ N e. NN0 ) -> ( A C N ) = { x e. ~P A | ( # ` x ) = N } )
14	5, 11, 13	3sstr4d 3648	1 |- ( ( A e. V /\ B C_ A /\ N e. NN0 ) -> ( B C N ) C_ ( A C N ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   ~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:  ramval  15712  ramub2  15718  ramub1lem2  15731