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Theorem mapprop 30907
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 457 . . . . . . 7  |-  ( ( X  e.  V  /\  A  e.  R )  ->  X  e.  V )
2 simpl 457 . . . . . . 7  |-  ( ( Y  e.  V  /\  B  e.  R )  ->  Y  e.  V )
31, 2anim12i 566 . . . . . 6  |-  ( ( ( X  e.  V  /\  A  e.  R
)  /\  ( Y  e.  V  /\  B  e.  R ) )  -> 
( X  e.  V  /\  Y  e.  V
) )
433adant3 1008 . . . . 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 461 . . . . . . 7  |-  ( ( X  e.  V  /\  A  e.  R )  ->  A  e.  R )
6 simpr 461 . . . . . . 7  |-  ( ( Y  e.  V  /\  B  e.  R )  ->  B  e.  R )
75, 6anim12i 566 . . . . . 6  |-  ( ( ( X  e.  V  /\  A  e.  R
)  /\  ( Y  e.  V  /\  B  e.  R ) )  -> 
( A  e.  R  /\  B  e.  R
) )
873adant3 1008 . . . . 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 457 . . . . . 6  |-  ( ( X  =/=  Y  /\  R  e.  W )  ->  X  =/=  Y )
1093ad2ant3 1011 . . . . 5  |-  ( ( ( X  e.  V  /\  A  e.  R
)  /\  ( Y  e.  V  /\  B  e.  R )  /\  ( X  =/=  Y  /\  R  e.  W ) )  ->  X  =/=  Y )
11 fprg 6003 . . . . 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 1219 . . . 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 5662 . . . 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 4140 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  R )  ->  { A ,  B }  C_  R )
177, 16syl 16 . . . 4  |-  ( ( ( X  e.  V  /\  A  e.  R
)  /\  ( Y  e.  V  /\  B  e.  R ) )  ->  { A ,  B }  C_  R )
18173adant3 1008 . . 3  |-  ( ( ( X  e.  V  /\  A  e.  R
)  /\  ( Y  e.  V  /\  B  e.  R )  /\  ( X  =/=  Y  /\  R  e.  W ) )  ->  { A ,  B }  C_  R )
19 fss 5678 . . 3  |-  ( ( F : { X ,  Y } --> { A ,  B }  /\  { A ,  B }  C_  R )  ->  F : { X ,  Y }
--> R )
2015, 18, 19syl2anc 661 . 2  |-  ( ( ( X  e.  V  /\  A  e.  R
)  /\  ( Y  e.  V  /\  B  e.  R )  /\  ( X  =/=  Y  /\  R  e.  W ) )  ->  F : { X ,  Y } --> R )
21 simpr 461 . . . 4  |-  ( ( X  =/=  Y  /\  R  e.  W )  ->  R  e.  W )
22213ad2ant3 1011 . . 3  |-  ( ( ( X  e.  V  /\  A  e.  R
)  /\  ( Y  e.  V  /\  B  e.  R )  /\  ( X  =/=  Y  /\  R  e.  W ) )  ->  R  e.  W )
23 prex 4645 . . 3  |-  { X ,  Y }  e.  _V
24 elmapg 7340 . . 3  |-  ( ( R  e.  W  /\  { X ,  Y }  e.  _V )  ->  ( F  e.  ( R  ^m  { X ,  Y } )  <->  F : { X ,  Y } --> R ) )
2522, 23, 24sylancl 662 . 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 ) )
2620, 25mpbird 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   _Vcvv 3078    C_ wss 3439   {cpr 3990   <.cop 3994   -->wf 5525  (class class class)co 6203    ^m cmap 7327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-map 7329
This theorem is referenced by:  lincvalpr  31107  ldepspr  31162
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