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Theorem pmss12g 7231
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.
StepHypRef Expression
1 xpss12 4940 . . . . . . 7  |-  ( ( B  C_  D  /\  A  C_  C )  -> 
( B  X.  A
)  C_  ( D  X.  C ) )
21ancoms 453 . . . . . 6  |-  ( ( A  C_  C  /\  B  C_  D )  -> 
( B  X.  A
)  C_  ( D  X.  C ) )
3 sstr 3359 . . . . . . 7  |-  ( ( f  C_  ( B  X.  A )  /\  ( B  X.  A )  C_  ( D  X.  C
) )  ->  f  C_  ( D  X.  C
) )
43expcom 435 . . . . . 6  |-  ( ( B  X.  A ) 
C_  ( D  X.  C )  ->  (
f  C_  ( B  X.  A )  ->  f  C_  ( D  X.  C
) ) )
52, 4syl 16 . . . . 5  |-  ( ( A  C_  C  /\  B  C_  D )  -> 
( f  C_  ( B  X.  A )  -> 
f  C_  ( D  X.  C ) ) )
65anim2d 565 . . . 4  |-  ( ( A  C_  C  /\  B  C_  D )  -> 
( ( Fun  f  /\  f  C_  ( B  X.  A ) )  ->  ( Fun  f  /\  f  C_  ( D  X.  C ) ) ) )
76adantr 465 . . 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 4433 . . . . 5  |-  ( ( A  C_  C  /\  C  e.  V )  ->  A  e.  _V )
9 ssexg 4433 . . . . 5  |-  ( ( B  C_  D  /\  D  e.  W )  ->  B  e.  _V )
10 elpmg 7220 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( f  e.  ( A  ^pm  B )  <->  ( Fun  f  /\  f  C_  ( B  X.  A
) ) ) )
118, 9, 10syl2an 477 . . . 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
) ) ) )
1211an4s 822 . . 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 7220 . . . 4  |-  ( ( C  e.  V  /\  D  e.  W )  ->  ( f  e.  ( C  ^pm  D )  <->  ( Fun  f  /\  f  C_  ( D  X.  C
) ) ) )
1413adantl 466 . . 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
) ) ) )
157, 12, 143imtr4d 268 . 2  |-  ( ( ( A  C_  C  /\  B  C_  D )  /\  ( C  e.  V  /\  D  e.  W ) )  -> 
( f  e.  ( A  ^pm  B )  ->  f  e.  ( C 
^pm  D ) ) )
1615ssrdv 3357 1  |-  ( ( ( A  C_  C  /\  B  C_  D )  /\  ( C  e.  V  /\  D  e.  W ) )  -> 
( A  ^pm  B
)  C_  ( C  ^pm  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1756   _Vcvv 2967    C_ wss 3323    X. cxp 4833   Fun wfun 5407  (class class class)co 6086    ^pm cpm 7207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-iota 5376  df-fun 5415  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-pm 7209
This theorem is referenced by:  lmres  18879  dvnadd  21378  caures  28609
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