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Theorem pmss12g 6794
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 4792 . . . . . . 7  |-  ( ( B  C_  D  /\  A  C_  C )  -> 
( B  X.  A
)  C_  ( D  X.  C ) )
21ancoms 439 . . . . . 6  |-  ( ( A  C_  C  /\  B  C_  D )  -> 
( B  X.  A
)  C_  ( D  X.  C ) )
3 sstr 3187 . . . . . . 7  |-  ( ( f  C_  ( B  X.  A )  /\  ( B  X.  A )  C_  ( D  X.  C
) )  ->  f  C_  ( D  X.  C
) )
43expcom 424 . . . . . 6  |-  ( ( B  X.  A ) 
C_  ( D  X.  C )  ->  (
f  C_  ( B  X.  A )  ->  f  C_  ( D  X.  C
) ) )
52, 4syl 15 . . . . 5  |-  ( ( A  C_  C  /\  B  C_  D )  -> 
( f  C_  ( B  X.  A )  -> 
f  C_  ( D  X.  C ) ) )
65anim2d 548 . . . 4  |-  ( ( A  C_  C  /\  B  C_  D )  -> 
( ( Fun  f  /\  f  C_  ( B  X.  A ) )  ->  ( Fun  f  /\  f  C_  ( D  X.  C ) ) ) )
76adantr 451 . . 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 4160 . . . . 5  |-  ( ( A  C_  C  /\  C  e.  V )  ->  A  e.  _V )
9 ssexg 4160 . . . . 5  |-  ( ( B  C_  D  /\  D  e.  W )  ->  B  e.  _V )
10 elpmg 6786 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( f  e.  ( A  ^pm  B )  <->  ( Fun  f  /\  f  C_  ( B  X.  A
) ) ) )
118, 9, 10syl2an 463 . . . 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 799 . . 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 6786 . . . 4  |-  ( ( C  e.  V  /\  D  e.  W )  ->  ( f  e.  ( C  ^pm  D )  <->  ( Fun  f  /\  f  C_  ( D  X.  C
) ) ) )
1413adantl 452 . . 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 259 . 2  |-  ( ( ( A  C_  C  /\  B  C_  D )  /\  ( C  e.  V  /\  D  e.  W ) )  -> 
( f  e.  ( A  ^pm  B )  ->  f  e.  ( C 
^pm  D ) ) )
1615ssrdv 3185 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 176    /\ wa 358    e. wcel 1684   _Vcvv 2788    C_ wss 3152    X. cxp 4687   Fun wfun 5249  (class class class)co 5858    ^pm cpm 6773
This theorem is referenced by:  lmres  17028  dvnadd  19278  caures  26476
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-pm 6775
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