Theorem funprg 4969

Description: A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.)

Assertion

Ref	Expression
funprg	|- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B ) -> Fun { <. A , C >. , <. B , D >. } )

Proof of Theorem funprg

Step	Hyp	Ref	Expression
1		simp1l 962	. . . 4 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B ) -> A e. V )
2		simp2l 964	. . . 4 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B ) -> C e. X )
3		funsng 4966	. . . 4 |- ( ( A e. V /\ C e. X ) -> Fun { <. A , C >. } )
4	1, 2, 3	syl2anc 403	. . 3 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B ) -> Fun { <. A , C >. } )
5		simp1r 963	. . . 4 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B ) -> B e. W )
6		simp2r 965	. . . 4 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B ) -> D e. Y )
7		funsng 4966	. . . 4 |- ( ( B e. W /\ D e. Y ) -> Fun { <. B , D >. } )
8	5, 6, 7	syl2anc 403	. . 3 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B ) -> Fun { <. B , D >. } )
9		dmsnopg 4812	. . . . . 6 |- ( C e. X -> dom { <. A , C >. } = { A } )
10	2, 9	syl 14	. . . . 5 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B ) -> dom { <. A , C >. } = { A } )
11		dmsnopg 4812	. . . . . 6 |- ( D e. Y -> dom { <. B , D >. } = { B } )
12	6, 11	syl 14	. . . . 5 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B ) -> dom { <. B , D >. } = { B } )
13	10, 12	ineq12d 3168	. . . 4 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B ) -> ( dom { <. A , C >. } i^i dom { <. B , D >. } ) = ( { A } i^i { B } ) )
14		disjsn2 3455	. . . . 5 |- ( A =/= B -> ( { A } i^i { B } ) = (/) )
15	14	3ad2ant3 961	. . . 4 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B ) -> ( { A } i^i { B } ) = (/) )
16	13, 15	eqtrd 2113	. . 3 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B ) -> ( dom { <. A , C >. } i^i dom { <. B , D >. } ) = (/) )
17		funun 4964	. . 3 |- ( ( ( Fun { <. A , C >. } /\ Fun { <. B , D >. } ) /\ ( dom { <. A , C >. } i^i dom { <. B , D >. } ) = (/) ) -> Fun ( { <. A , C >. } u. { <. B , D >. } ) )
18	4, 8, 16, 17	syl21anc 1168	. 2 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B ) -> Fun ( { <. A , C >. } u. { <. B , D >. } ) )
19		df-pr 3405	. . 3 |- { <. A , C >. , <. B , D >. } = ( { <. A , C >. } u. { <. B , D >. } )
20	19	funeqi 4942	. 2 |- ( Fun { <. A , C >. , <. B , D >. } <-> Fun ( { <. A , C >. } u. { <. B , D >. } ) )
21	18, 20	sylibr 132	1 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B ) -> Fun { <. A , C >. , <. B , D >. } )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   = wceq 1284   e. wcel 1433   =/= wne 2245   u. cun 2971   i^i cin 2972   (/)c0 3251   {csn 3398   {cpr 3399   <.cop 3401   dom cdm 4363   Fun wfun 4916

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-fun 4924

This theorem is referenced by:  funtpg  4970  funpr  4971  fnprg  4974