Theorem pmss12g 6922

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 4862	. . . . . . 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 3250	. . . . . . 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 4254	. . . . 5 |- ( ( A C_ C /\ C e. V ) -> A e. _V )
9		ssexg 4254	. . . . 5 |- ( ( B C_ D /\ D e. W ) -> B e. _V )
10		elpmg 6911	. . . . 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 592	. . 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 6911	. . . 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 3248	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 2205   _Vcvv 2815   C_ wss 3214   X. cxp 4752   Fun wfun 5351  (class class class)co 6058   ^pm cpm 6896

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pm 6898

This theorem is referenced by:  lmres  15239  dvidsslem  15684