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Theorem pmss12g 6675
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 4734 . . . . . . 7  |-  ( ( B  C_  D  /\  A  C_  C )  -> 
( B  X.  A
)  C_  ( D  X.  C ) )
21ancoms 268 . . . . . 6  |-  ( ( A  C_  C  /\  B  C_  D )  -> 
( B  X.  A
)  C_  ( D  X.  C ) )
3 sstr 3164 . . . . . . 7  |-  ( ( f  C_  ( B  X.  A )  /\  ( B  X.  A )  C_  ( D  X.  C
) )  ->  f  C_  ( D  X.  C
) )
43expcom 116 . . . . . 6  |-  ( ( B  X.  A ) 
C_  ( D  X.  C )  ->  (
f  C_  ( B  X.  A )  ->  f  C_  ( D  X.  C
) ) )
52, 4syl 14 . . . . 5  |-  ( ( A  C_  C  /\  B  C_  D )  -> 
( f  C_  ( B  X.  A )  -> 
f  C_  ( D  X.  C ) ) )
65anim2d 337 . . . 4  |-  ( ( A  C_  C  /\  B  C_  D )  -> 
( ( Fun  f  /\  f  C_  ( B  X.  A ) )  ->  ( Fun  f  /\  f  C_  ( D  X.  C ) ) ) )
76adantr 276 . . 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 4143 . . . . 5  |-  ( ( A  C_  C  /\  C  e.  V )  ->  A  e.  _V )
9 ssexg 4143 . . . . 5  |-  ( ( B  C_  D  /\  D  e.  W )  ->  B  e.  _V )
10 elpmg 6664 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( f  e.  ( A  ^pm  B )  <->  ( Fun  f  /\  f  C_  ( B  X.  A
) ) ) )
118, 9, 10syl2an 289 . . . 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 588 . . 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 6664 . . . 4  |-  ( ( C  e.  V  /\  D  e.  W )  ->  ( f  e.  ( C  ^pm  D )  <->  ( Fun  f  /\  f  C_  ( D  X.  C
) ) ) )
1413adantl 277 . . 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 203 . 2  |-  ( ( ( A  C_  C  /\  B  C_  D )  /\  ( C  e.  V  /\  D  e.  W ) )  -> 
( f  e.  ( A  ^pm  B )  ->  f  e.  ( C 
^pm  D ) ) )
1615ssrdv 3162 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    /\ wa 104    <-> wb 105    e. wcel 2148   _Vcvv 2738    C_ wss 3130    X. cxp 4625   Fun wfun 5211  (class class class)co 5875    ^pm cpm 6649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pm 6651
This theorem is referenced by:  lmres  13751
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