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Theorem pmss12g 6999
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 440 . . . . . 6  |-  ( ( A  C_  C  /\  B  C_  D )  -> 
( B  X.  A
)  C_  ( D  X.  C ) )
3 sstr 3316 . . . . . . 7  |-  ( ( f  C_  ( B  X.  A )  /\  ( B  X.  A )  C_  ( D  X.  C
) )  ->  f  C_  ( D  X.  C
) )
43expcom 425 . . . . . 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 549 . . . 4  |-  ( ( A  C_  C  /\  B  C_  D )  -> 
( ( Fun  f  /\  f  C_  ( B  X.  A ) )  ->  ( Fun  f  /\  f  C_  ( D  X.  C ) ) ) )
76adantr 452 . . 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 4309 . . . . 5  |-  ( ( A  C_  C  /\  C  e.  V )  ->  A  e.  _V )
9 ssexg 4309 . . . . 5  |-  ( ( B  C_  D  /\  D  e.  W )  ->  B  e.  _V )
10 elpmg 6991 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( f  e.  ( A  ^pm  B )  <->  ( Fun  f  /\  f  C_  ( B  X.  A
) ) ) )
118, 9, 10syl2an 464 . . . 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 800 . . 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 6991 . . . 4  |-  ( ( C  e.  V  /\  D  e.  W )  ->  ( f  e.  ( C  ^pm  D )  <->  ( Fun  f  /\  f  C_  ( D  X.  C
) ) ) )
1413adantl 453 . . 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 260 . 2  |-  ( ( ( A  C_  C  /\  B  C_  D )  /\  ( C  e.  V  /\  D  e.  W ) )  -> 
( f  e.  ( A  ^pm  B )  ->  f  e.  ( C 
^pm  D ) ) )
1615ssrdv 3314 1  |-  ( ( ( A  C_  C  /\  B  C_  D )  /\  ( C  e.  V  /\  D  e.  W ) )  -> 
( A  ^pm  B
)  C_  ( C  ^pm  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1721   _Vcvv 2916    C_ wss 3280    X. cxp 4835   Fun wfun 5407  (class class class)co 6040    ^pm cpm 6978
This theorem is referenced by:  lmres  17318  dvnadd  19768  caures  26356
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-pm 6980
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