Theorem mapprop 42124

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 473	. . . . . . 7 |- ( ( X e. V /\ A e. R ) -> X e. V )
2		simpl 473	. . . . . . 7 |- ( ( Y e. V /\ B e. R ) -> Y e. V )
3	1, 2	anim12i 590	. . . . . 6 |- ( ( ( X e. V /\ A e. R ) /\ ( Y e. V /\ B e. R ) ) -> ( X e. V /\ Y e. V ) )
4	3	3adant3 1081	. . . . 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 477	. . . . . . 7 |- ( ( X e. V /\ A e. R ) -> A e. R )
6		simpr 477	. . . . . . 7 |- ( ( Y e. V /\ B e. R ) -> B e. R )
7	5, 6	anim12i 590	. . . . . 6 |- ( ( ( X e. V /\ A e. R ) /\ ( Y e. V /\ B e. R ) ) -> ( A e. R /\ B e. R ) )
8	7	3adant3 1081	. . . . 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 473	. . . . . 6 |- ( ( X =/= Y /\ R e. W ) -> X =/= Y )
10	9	3ad2ant3 1084	. . . . 5 |- ( ( ( X e. V /\ A e. R ) /\ ( Y e. V /\ B e. R ) /\ ( X =/= Y /\ R e. W ) ) -> X =/= Y )
11		fprg 6422	. . . . 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 1326	. . . 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 6036	. . . 4 |- ( F : { X , Y } --> { A , B } <-> { <. X , A >. , <. Y , B >. } : { X , Y } --> { A , B } )
15	12, 14	sylibr 224	. . 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 4353	. . . . 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 1081	. . 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 6057	. 2 |- ( ( ( X e. V /\ A e. R ) /\ ( Y e. V /\ B e. R ) /\ ( X =/= Y /\ R e. W ) ) -> F : { X , Y } --> R )
20		simpr 477	. . . 4 |- ( ( X =/= Y /\ R e. W ) -> R e. W )
21	20	3ad2ant3 1084	. . 3 |- ( ( ( X e. V /\ A e. R ) /\ ( Y e. V /\ B e. R ) /\ ( X =/= Y /\ R e. W ) ) -> R e. W )
22		prex 4909	. . 3 |- { X , Y } e. _V
23		elmapg 7870	. . 3 |- ( ( R e. W /\ { X , Y } e. _V ) -> ( F e. ( R ^m { X , Y } ) <-> F : { X , Y } --> R ) )
24	21, 22, 23	sylancl 694	. 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 247	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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   {cpr 4179   <.cop 4183   -->wf 5884  (class class class)co 6650   ^m cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859

This theorem is referenced by:  lincvalpr  42207  ldepspr  42262