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

Step	Hyp	Ref	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 )
3	1, 2	anim12i 566	. . . . . 6 |- ( ( ( X e. V /\ A e. R ) /\ ( Y e. V /\ B e. R ) ) -> ( X e. V /\ Y e. V ) )
4	3	3adant3 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 )
7	5, 6	anim12i 566	. . . . . 6 |- ( ( ( X e. V /\ A e. R ) /\ ( Y e. V /\ B e. R ) ) -> ( A e. R /\ B e. R ) )
8	7	3adant3 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 )
10	9	3ad2ant3 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 } )
12	4, 8, 10, 11	syl3anc 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 >. }
14	13	feq1i 5662	. . . 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 4140	. . . . 5 |- ( ( A e. R /\ B e. R ) -> { A , B } C_ R )
17	7, 16	syl 16	. . . 4 |- ( ( ( X e. V /\ A e. R ) /\ ( Y e. V /\ B e. R ) ) -> { A , B } C_ R )
18	17	3adant3 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 )
20	15, 18, 19	syl2anc 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 )
22	21	3ad2ant3 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 ) )
25	22, 23, 24	sylancl 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 ) )
26	20, 25	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 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