Theorem funtpg 5942

Description: A set of three pairs is a function if their first members are different. (Contributed by Alexander van der Vekens, 5-Dec-2017.) (Proof shortened by JJ, 14-Jul-2021.)

Assertion

Ref	Expression
funtpg	|- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ ( A e. F /\ B e. G /\ C e. H ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> Fun { <. X , A >. , <. Y , B >. , <. Z , C >. } )

Proof of Theorem funtpg

Step	Hyp	Ref	Expression
1		3simpa 1058	. . . 4 |- ( ( X e. U /\ Y e. V /\ Z e. W ) -> ( X e. U /\ Y e. V ) )
2		3simpa 1058	. . . 4 |- ( ( A e. F /\ B e. G /\ C e. H ) -> ( A e. F /\ B e. G ) )
3		simp1 1061	. . . 4 |- ( ( X =/= Y /\ X =/= Z /\ Y =/= Z ) -> X =/= Y )
4		funprg 5940	. . . 4 |- ( ( ( X e. U /\ Y e. V ) /\ ( A e. F /\ B e. G ) /\ X =/= Y ) -> Fun { <. X , A >. , <. Y , B >. } )
5	1, 2, 3, 4	syl3an 1368	. . 3 |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ ( A e. F /\ B e. G /\ C e. H ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> Fun { <. X , A >. , <. Y , B >. } )
6		simp3 1063	. . . . 5 |- ( ( X e. U /\ Y e. V /\ Z e. W ) -> Z e. W )
7		simp3 1063	. . . . 5 |- ( ( A e. F /\ B e. G /\ C e. H ) -> C e. H )
8		funsng 5937	. . . . 5 |- ( ( Z e. W /\ C e. H ) -> Fun { <. Z , C >. } )
9	6, 7, 8	syl2an 494	. . . 4 |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ ( A e. F /\ B e. G /\ C e. H ) ) -> Fun { <. Z , C >. } )
10	9	3adant3 1081	. . 3 |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ ( A e. F /\ B e. G /\ C e. H ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> Fun { <. Z , C >. } )
11		dmpropg 5608	. . . . . . 7 |- ( ( A e. F /\ B e. G ) -> dom { <. X , A >. , <. Y , B >. } = { X , Y } )
12		dmsnopg 5606	. . . . . . 7 |- ( C e. H -> dom { <. Z , C >. } = { Z } )
13	11, 12	ineqan12d 3816	. . . . . 6 |- ( ( ( A e. F /\ B e. G ) /\ C e. H ) -> ( dom { <. X , A >. , <. Y , B >. } i^i dom { <. Z , C >. } ) = ( { X , Y } i^i { Z } ) )
14	13	3impa 1259	. . . . 5 |- ( ( A e. F /\ B e. G /\ C e. H ) -> ( dom { <. X , A >. , <. Y , B >. } i^i dom { <. Z , C >. } ) = ( { X , Y } i^i { Z } ) )
15		disjprsn 4250	. . . . . 6 |- ( ( X =/= Z /\ Y =/= Z ) -> ( { X , Y } i^i { Z } ) = (/) )
16	15	3adant1 1079	. . . . 5 |- ( ( X =/= Y /\ X =/= Z /\ Y =/= Z ) -> ( { X , Y } i^i { Z } ) = (/) )
17	14, 16	sylan9eq 2676	. . . 4 |- ( ( ( A e. F /\ B e. G /\ C e. H ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( dom { <. X , A >. , <. Y , B >. } i^i dom { <. Z , C >. } ) = (/) )
18	17	3adant1 1079	. . 3 |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ ( A e. F /\ B e. G /\ C e. H ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( dom { <. X , A >. , <. Y , B >. } i^i dom { <. Z , C >. } ) = (/) )
19		funun 5932	. . 3 |- ( ( ( Fun { <. X , A >. , <. Y , B >. } /\ Fun { <. Z , C >. } ) /\ ( dom { <. X , A >. , <. Y , B >. } i^i dom { <. Z , C >. } ) = (/) ) -> Fun ( { <. X , A >. , <. Y , B >. } u. { <. Z , C >. } ) )
20	5, 10, 18, 19	syl21anc 1325	. 2 |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ ( A e. F /\ B e. G /\ C e. H ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> Fun ( { <. X , A >. , <. Y , B >. } u. { <. Z , C >. } ) )
21		df-tp 4182	. . 3 |- { <. X , A >. , <. Y , B >. , <. Z , C >. } = ( { <. X , A >. , <. Y , B >. } u. { <. Z , C >. } )
22	21	funeqi 5909	. 2 |- ( Fun { <. X , A >. , <. Y , B >. , <. Z , C >. } <-> Fun ( { <. X , A >. , <. Y , B >. } u. { <. Z , C >. } ) )
23	20, 22	sylibr 224	1 |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ ( A e. F /\ B e. G /\ C e. H ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> Fun { <. X , A >. , <. Y , B >. , <. Z , C >. } )

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   {ctp 4181   <.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-tp 4182  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:  fntpg  5948  estrres  16779