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

Step	Hyp	Ref	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 )
3	1, 2	anim12i 564	. . . . . 6 |- ( ( ( X e. V /\ A e. R ) /\ ( Y e. V /\ B e. R ) ) -> ( X e. V /\ Y e. V ) )
4	3	3adant3 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 )
7	5, 6	anim12i 564	. . . . . 6 |- ( ( ( X e. V /\ A e. R ) /\ ( Y e. V /\ B e. R ) ) -> ( A e. R /\ B e. R ) )
8	7	3adant3 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 )
10	9	3ad2ant3 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 } )
12	4, 8, 10, 11	syl3anc 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 >. }
14	13	feq1i 5705	. . . 4 |- ( F : { X , Y } --> { A , B } <-> { <. X , A >. , <. Y , B >. } : { X , Y } --> { A , B } )
15	12, 14	sylibr 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 )
17	7, 16	syl 17	. . . 4 |- ( ( ( X e. V /\ A e. R ) /\ ( Y e. V /\ B e. R ) ) -> { A , B } C_ R )
18	17	3adant3 1017	. . 3 |- ( ( ( X e. V /\ A e. R ) /\ ( Y e. V /\ B e. R ) /\ ( X =/= Y /\ R e. W ) ) -> { A , B } C_ R )
19	15, 18	fssd 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 )
21	20	3ad2ant3 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 ) )
24	21, 22, 23	sylancl 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 ) )
25	19, 24	mpbird 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 } ) )

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