Theorem pmss12g 6830

Description: Subset relation for the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)

Assertion

Ref	Expression
pmss12g	|- ( ( ( A C_ C /\ B C_ D ) /\ ( C e. V /\ D e. W ) ) -> ( A ^pm B ) C_ ( C ^pm D ) )

Proof of Theorem pmss12g

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpss12 4826	. . . . . . 7 |- ( ( B C_ D /\ A C_ C ) -> ( B X. A ) C_ ( D X. C ) )
2	1	ancoms 268	. . . . . 6 |- ( ( A C_ C /\ B C_ D ) -> ( B X. A ) C_ ( D X. C ) )
3		sstr 3232	. . . . . . 7 |- ( ( f C_ ( B X. A ) /\ ( B X. A ) C_ ( D X. C ) ) -> f C_ ( D X. C ) )
4	3	expcom 116	. . . . . 6 |- ( ( B X. A ) C_ ( D X. C ) -> ( f C_ ( B X. A ) -> f C_ ( D X. C ) ) )
5	2, 4	syl 14	. . . . 5 |- ( ( A C_ C /\ B C_ D ) -> ( f C_ ( B X. A ) -> f C_ ( D X. C ) ) )
6	5	anim2d 337	. . . 4 |- ( ( A C_ C /\ B C_ D ) -> ( ( Fun f /\ f C_ ( B X. A ) ) -> ( Fun f /\ f C_ ( D X. C ) ) ) )
7	6	adantr 276	. . 3 |- ( ( ( A C_ C /\ B C_ D ) /\ ( C e. V /\ D e. W ) ) -> ( ( Fun f /\ f C_ ( B X. A ) ) -> ( Fun f /\ f C_ ( D X. C ) ) ) )
8		ssexg 4223	. . . . 5 |- ( ( A C_ C /\ C e. V ) -> A e. _V )
9		ssexg 4223	. . . . 5 |- ( ( B C_ D /\ D e. W ) -> B e. _V )
10		elpmg 6819	. . . . 5 |- ( ( A e. _V /\ B e. _V ) -> ( f e. ( A ^pm B ) <-> ( Fun f /\ f C_ ( B X. A ) ) ) )
11	8, 9, 10	syl2an 289	. . . 4 |- ( ( ( A C_ C /\ C e. V ) /\ ( B C_ D /\ D e. W ) ) -> ( f e. ( A ^pm B ) <-> ( Fun f /\ f C_ ( B X. A ) ) ) )
12	11	an4s 590	. . 3 |- ( ( ( A C_ C /\ B C_ D ) /\ ( C e. V /\ D e. W ) ) -> ( f e. ( A ^pm B ) <-> ( Fun f /\ f C_ ( B X. A ) ) ) )
13		elpmg 6819	. . . 4 |- ( ( C e. V /\ D e. W ) -> ( f e. ( C ^pm D ) <-> ( Fun f /\ f C_ ( D X. C ) ) ) )
14	13	adantl 277	. . 3 |- ( ( ( A C_ C /\ B C_ D ) /\ ( C e. V /\ D e. W ) ) -> ( f e. ( C ^pm D ) <-> ( Fun f /\ f C_ ( D X. C ) ) ) )
15	7, 12, 14	3imtr4d 203	. 2 |- ( ( ( A C_ C /\ B C_ D ) /\ ( C e. V /\ D e. W ) ) -> ( f e. ( A ^pm B ) -> f e. ( C ^pm D ) ) )
16	15	ssrdv 3230	1 |- ( ( ( A C_ C /\ B C_ D ) /\ ( C e. V /\ D e. W ) ) -> ( A ^pm B ) C_ ( C ^pm D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2200   _Vcvv 2799   C_ wss 3197   X. cxp 4717   Fun wfun 5312  (class class class)co 6007   ^pm cpm 6804

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 617  ax-in2 618  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-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  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-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pm 6806

This theorem is referenced by:  lmres  14937  dvidsslem  15382