Theorem pmss12g 6632

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 4705	. . . . . . 7 |- ( ( B C_ D /\ A C_ C ) -> ( B X. A ) C_ ( D X. C ) )
2	1	ancoms 266	. . . . . 6 |- ( ( A C_ C /\ B C_ D ) -> ( B X. A ) C_ ( D X. C ) )
3		sstr 3145	. . . . . . 7 |- ( ( f C_ ( B X. A ) /\ ( B X. A ) C_ ( D X. C ) ) -> f C_ ( D X. C ) )
4	3	expcom 115	. . . . . 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 335	. . . 4 |- ( ( A C_ C /\ B C_ D ) -> ( ( Fun f /\ f C_ ( B X. A ) ) -> ( Fun f /\ f C_ ( D X. C ) ) ) )
7	6	adantr 274	. . 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 4115	. . . . 5 |- ( ( A C_ C /\ C e. V ) -> A e. _V )
9		ssexg 4115	. . . . 5 |- ( ( B C_ D /\ D e. W ) -> B e. _V )
10		elpmg 6621	. . . . 5 |- ( ( A e. _V /\ B e. _V ) -> ( f e. ( A ^pm B ) <-> ( Fun f /\ f C_ ( B X. A ) ) ) )
11	8, 9, 10	syl2an 287	. . . 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 578	. . 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 6621	. . . 4 |- ( ( C e. V /\ D e. W ) -> ( f e. ( C ^pm D ) <-> ( Fun f /\ f C_ ( D X. C ) ) ) )
14	13	adantl 275	. . 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 202	. 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 3143	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 103   <-> wb 104   e. wcel 2135   _Vcvv 2721   C_ wss 3111   X. cxp 4596   Fun wfun 5176  (class class class)co 5836   ^pm cpm 6606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-sbc 2947  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-iota 5147  df-fun 5184  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-pm 6608

This theorem is referenced by:  lmres  12789