Theorem pmss12g 6569

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 4646	. . . . . . 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 3105	. . . . . . 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 4067	. . . . 5 |- ( ( A C_ C /\ C e. V ) -> A e. _V )
9		ssexg 4067	. . . . 5 |- ( ( B C_ D /\ D e. W ) -> B e. _V )
10		elpmg 6558	. . . . 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 577	. . 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 6558	. . . 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 3103	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 1480   _Vcvv 2686   C_ wss 3071   X. cxp 4537   Fun wfun 5117  (class class class)co 5774   ^pm cpm 6543

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pm 6545

This theorem is referenced by:  lmres  12417