Theorem funprg 5940

Description: A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.)

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		funsng 5937	. . . . . 6 |- ( ( A e. V /\ C e. X ) -> Fun { <. A , C >. } )
2		funsng 5937	. . . . . 6 |- ( ( B e. W /\ D e. Y ) -> Fun { <. B , D >. } )
3	1, 2	anim12i 590	. . . . 5 |- ( ( ( A e. V /\ C e. X ) /\ ( B e. W /\ D e. Y ) ) -> ( Fun { <. A , C >. } /\ Fun { <. B , D >. } ) )
4	3	an4s 869	. . . 4 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) -> ( Fun { <. A , C >. } /\ Fun { <. B , D >. } ) )
5	4	3adant3 1081	. . 3 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B ) -> ( Fun { <. A , C >. } /\ Fun { <. B , D >. } ) )
6		dmsnopg 5606	. . . . . 6 |- ( C e. X -> dom { <. A , C >. } = { A } )
7		dmsnopg 5606	. . . . . 6 |- ( D e. Y -> dom { <. B , D >. } = { B } )
8	6, 7	ineqan12d 3816	. . . . 5 |- ( ( C e. X /\ D e. Y ) -> ( dom { <. A , C >. } i^i dom { <. B , D >. } ) = ( { A } i^i { B } ) )
9		disjsn2 4247	. . . . 5 |- ( A =/= B -> ( { A } i^i { B } ) = (/) )
10	8, 9	sylan9eq 2676	. . . 4 |- ( ( ( C e. X /\ D e. Y ) /\ A =/= B ) -> ( dom { <. A , C >. } i^i dom { <. B , D >. } ) = (/) )
11	10	3adant1 1079	. . 3 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B ) -> ( dom { <. A , C >. } i^i dom { <. B , D >. } ) = (/) )
12		funun 5932	. . 3 |- ( ( ( Fun { <. A , C >. } /\ Fun { <. B , D >. } ) /\ ( dom { <. A , C >. } i^i dom { <. B , D >. } ) = (/) ) -> Fun ( { <. A , C >. } u. { <. B , D >. } ) )
13	5, 11, 12	syl2anc 693	. 2 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B ) -> Fun ( { <. A , C >. } u. { <. B , D >. } ) )
14		df-pr 4180	. . 3 |- { <. A , C >. , <. B , D >. } = ( { <. A , C >. } u. { <. B , D >. } )
15	14	funeqi 5909	. 2 |- ( Fun { <. A , C >. , <. B , D >. } <-> Fun ( { <. A , C >. } u. { <. B , D >. } ) )
16	13, 15	sylibr 224	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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   {cpr 4179   <.cop 4183   dom cdm 5114   Fun wfun 5882

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  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-fun 5890

This theorem is referenced by:  funtpg  5942  funtpgOLD  5943  funpr  5944  fnprg  5947  fpropnf1  6524