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Theorem mapprop 38427
Description: An unordered pair containing two ordered pairs as an element of the mapping operation. (Contributed by AV, 16-Apr-2019.)
Hypothesis
Ref Expression
mapprop.f  |-  F  =  { <. X ,  A >. ,  <. Y ,  B >. }
Assertion
Ref Expression
mapprop  |-  ( ( ( X  e.  V  /\  A  e.  R
)  /\  ( Y  e.  V  /\  B  e.  R )  /\  ( X  =/=  Y  /\  R  e.  W ) )  ->  F  e.  ( R  ^m  { X ,  Y } ) )

Proof of Theorem mapprop
StepHypRef Expression
1 simpl 455 . . . . . . 7  |-  ( ( X  e.  V  /\  A  e.  R )  ->  X  e.  V )
2 simpl 455 . . . . . . 7  |-  ( ( Y  e.  V  /\  B  e.  R )  ->  Y  e.  V )
31, 2anim12i 564 . . . . . 6  |-  ( ( ( X  e.  V  /\  A  e.  R
)  /\  ( Y  e.  V  /\  B  e.  R ) )  -> 
( X  e.  V  /\  Y  e.  V
) )
433adant3 1017 . . . . 5  |-  ( ( ( X  e.  V  /\  A  e.  R
)  /\  ( Y  e.  V  /\  B  e.  R )  /\  ( X  =/=  Y  /\  R  e.  W ) )  -> 
( X  e.  V  /\  Y  e.  V
) )
5 simpr 459 . . . . . . 7  |-  ( ( X  e.  V  /\  A  e.  R )  ->  A  e.  R )
6 simpr 459 . . . . . . 7  |-  ( ( Y  e.  V  /\  B  e.  R )  ->  B  e.  R )
75, 6anim12i 564 . . . . . 6  |-  ( ( ( X  e.  V  /\  A  e.  R
)  /\  ( Y  e.  V  /\  B  e.  R ) )  -> 
( A  e.  R  /\  B  e.  R
) )
873adant3 1017 . . . . 5  |-  ( ( ( X  e.  V  /\  A  e.  R
)  /\  ( Y  e.  V  /\  B  e.  R )  /\  ( X  =/=  Y  /\  R  e.  W ) )  -> 
( A  e.  R  /\  B  e.  R
) )
9 simpl 455 . . . . . 6  |-  ( ( X  =/=  Y  /\  R  e.  W )  ->  X  =/=  Y )
1093ad2ant3 1020 . . . . 5  |-  ( ( ( X  e.  V  /\  A  e.  R
)  /\  ( Y  e.  V  /\  B  e.  R )  /\  ( X  =/=  Y  /\  R  e.  W ) )  ->  X  =/=  Y )
11 fprg 6059 . . . . 5  |-  ( ( ( X  e.  V  /\  Y  e.  V
)  /\  ( A  e.  R  /\  B  e.  R )  /\  X  =/=  Y )  ->  { <. X ,  A >. ,  <. Y ,  B >. } : { X ,  Y } --> { A ,  B }
)
124, 8, 10, 11syl3anc 1230 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  R
)  /\  ( Y  e.  V  /\  B  e.  R )  /\  ( X  =/=  Y  /\  R  e.  W ) )  ->  { <. X ,  A >. ,  <. Y ,  B >. } : { X ,  Y } --> { A ,  B } )
13 mapprop.f . . . . 5  |-  F  =  { <. X ,  A >. ,  <. Y ,  B >. }
1413feq1i 5705 . . . 4  |-  ( F : { X ,  Y } --> { A ,  B }  <->  { <. X ,  A >. ,  <. Y ,  B >. } : { X ,  Y } --> { A ,  B } )
1512, 14sylibr 212 . . 3  |-  ( ( ( X  e.  V  /\  A  e.  R
)  /\  ( Y  e.  V  /\  B  e.  R )  /\  ( X  =/=  Y  /\  R  e.  W ) )  ->  F : { X ,  Y } --> { A ,  B } )
16 prssi 4127 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  R )  ->  { A ,  B }  C_  R )
177, 16syl 17 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  R
)  /\  ( Y  e.  V  /\  B  e.  R ) )  ->  { A ,  B }  C_  R )
18173adant3 1017 . . 3  |-  ( ( ( X  e.  V  /\  A  e.  R
)  /\  ( Y  e.  V  /\  B  e.  R )  /\  ( X  =/=  Y  /\  R  e.  W ) )  ->  { A ,  B }  C_  R )
1915, 18fssd 5722 . 2  |-  ( ( ( X  e.  V  /\  A  e.  R
)  /\  ( Y  e.  V  /\  B  e.  R )  /\  ( X  =/=  Y  /\  R  e.  W ) )  ->  F : { X ,  Y } --> R )
20 simpr 459 . . . 4  |-  ( ( X  =/=  Y  /\  R  e.  W )  ->  R  e.  W )
21203ad2ant3 1020 . . 3  |-  ( ( ( X  e.  V  /\  A  e.  R
)  /\  ( Y  e.  V  /\  B  e.  R )  /\  ( X  =/=  Y  /\  R  e.  W ) )  ->  R  e.  W )
22 prex 4632 . . 3  |-  { X ,  Y }  e.  _V
23 elmapg 7469 . . 3  |-  ( ( R  e.  W  /\  { X ,  Y }  e.  _V )  ->  ( F  e.  ( R  ^m  { X ,  Y } )  <->  F : { X ,  Y } --> R ) )
2421, 22, 23sylancl 660 . 2  |-  ( ( ( X  e.  V  /\  A  e.  R
)  /\  ( Y  e.  V  /\  B  e.  R )  /\  ( X  =/=  Y  /\  R  e.  W ) )  -> 
( F  e.  ( R  ^m  { X ,  Y } )  <->  F : { X ,  Y } --> R ) )
2519, 24mpbird 232 1  |-  ( ( ( X  e.  V  /\  A  e.  R
)  /\  ( Y  e.  V  /\  B  e.  R )  /\  ( X  =/=  Y  /\  R  e.  W ) )  ->  F  e.  ( R  ^m  { X ,  Y } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   _Vcvv 3058    C_ wss 3413   {cpr 3973   <.cop 3977   -->wf 5564  (class class class)co 6277    ^m cmap 7456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7458
This theorem is referenced by:  lincvalpr  38511  ldepspr  38566
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