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Theorem pmss12g 7040
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 4981 . . . . . . 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 3356 . . . . . . 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 4349 . . . . 5  |-  ( ( A  C_  C  /\  C  e.  V )  ->  A  e.  _V )
9 ssexg 4349 . . . . 5  |-  ( ( B  C_  D  /\  D  e.  W )  ->  B  e.  _V )
10 elpmg 7032 . . . . 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 7032 . . . 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 3354 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 1725   _Vcvv 2956    C_ wss 3320    X. cxp 4876   Fun wfun 5448  (class class class)co 6081    ^pm cpm 7019
This theorem is referenced by:  lmres  17364  dvnadd  19815  caures  26466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-pm 7021
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