Theorem pmss12g 6762

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 4782	. . . . . . 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 3201	. . . . . . 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 4183	. . . . 5 |- ( ( A C_ C /\ C e. V ) -> A e. _V )
9		ssexg 4183	. . . . 5 |- ( ( B C_ D /\ D e. W ) -> B e. _V )
10		elpmg 6751	. . . . 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 588	. . 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 6751	. . . 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 3199	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 2176   _Vcvv 2772   C_ wss 3166   X. cxp 4673   Fun wfun 5265  (class class class)co 5944   ^pm cpm 6736

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pm 6738

This theorem is referenced by:  lmres  14720  dvidsslem  15165