Theorem suppsnop 6917

Description: The support of a singleton of an ordered pair. (Contributed by AV, 12-Apr-2019.)

Hypothesis

Ref	Expression
suppsnop.f	|- F = { <. X , Y >. }

Assertion

Ref	Expression
suppsnop	|- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( F supp Z ) = if ( Y = Z , (/) , { X } ) )

Proof of Theorem suppsnop

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1osng 5844	. . . . . . 7 |- ( ( X e. V /\ Y e. W ) -> { <. X , Y >. } : { X } -1-1-onto-> { Y } )
2		f1of 5806	. . . . . . 7 |- ( { <. X , Y >. } : { X } -1-1-onto-> { Y } -> { <. X , Y >. } : { X } --> { Y } )
3	1, 2	syl 16	. . . . . 6 |- ( ( X e. V /\ Y e. W ) -> { <. X , Y >. } : { X } --> { Y } )
4	3	3adant3 1017	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { <. X , Y >. } : { X } --> { Y } )
5		suppsnop.f	. . . . . 6 |- F = { <. X , Y >. }
6	5	feq1i 5713	. . . . 5 |- ( F : { X } --> { Y } <-> { <. X , Y >. } : { X } --> { Y } )
7	4, 6	sylibr 212	. . . 4 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> F : { X } --> { Y } )
8		snex 4678	. . . . 5 |- { X } e. _V
9	8	a1i 11	. . . 4 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { X } e. _V )
10		fex 6130	. . . 4 |- ( ( F : { X } --> { Y } /\ { X } e. _V ) -> F e. _V )
11	7, 9, 10	syl2anc 661	. . 3 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> F e. _V )
12		simp3 999	. . 3 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> Z e. U )
13		suppval 6905	. . 3 |- ( ( F e. _V /\ Z e. U ) -> ( F supp Z ) = { x e. dom F | ( F " { x } ) =/= { Z } } )
14	11, 12, 13	syl2anc 661	. 2 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( F supp Z ) = { x e. dom F | ( F " { x } ) =/= { Z } } )
15	5	a1i 11	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> F = { <. X , Y >. } )
16	15	dmeqd 5195	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> dom F = dom { <. X , Y >. } )
17		dmsnopg 5469	. . . . . 6 |- ( Y e. W -> dom { <. X , Y >. } = { X } )
18	17	3ad2ant2 1019	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> dom { <. X , Y >. } = { X } )
19	16, 18	eqtrd 2484	. . . 4 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> dom F = { X } )
20		rabeq 3089	. . . 4 |- ( dom F = { X } -> { x e. dom F | ( F " { x } ) =/= { Z } } = { x e. { X } | ( F " { x } ) =/= { Z } } )
21	19, 20	syl 16	. . 3 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { x e. dom F | ( F " { x } ) =/= { Z } } = { x e. { X } | ( F " { x } ) =/= { Z } } )
22		sneq 4024	. . . . . 6 |- ( x = X -> { x } = { X } )
23	22	imaeq2d 5327	. . . . 5 |- ( x = X -> ( F " { x } ) = ( F " { X } ) )
24	23	neeq1d 2720	. . . 4 |- ( x = X -> ( ( F " { x } ) =/= { Z } <-> ( F " { X } ) =/= { Z } ) )
25	24	rabsnif 4084	. . 3 |- { x e. { X } | ( F " { x } ) =/= { Z } } = if ( ( F " { X } ) =/= { Z } , { X } , (/) )
26	21, 25	syl6eq 2500	. 2 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { x e. dom F | ( F " { x } ) =/= { Z } } = if ( ( F " { X } ) =/= { Z } , { X } , (/) ) )
27		fnsng 5625	. . . . . . . . 9 |- ( ( X e. V /\ Y e. W ) -> { <. X , Y >. } Fn { X } )
28	27	3adant3 1017	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { <. X , Y >. } Fn { X } )
29	5	eqcomi 2456	. . . . . . . . . 10 |- { <. X , Y >. } = F
30	29	a1i 11	. . . . . . . . 9 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { <. X , Y >. } = F )
31	30	fneq1d 5661	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { <. X , Y >. } Fn { X } <-> F Fn { X } ) )
32	28, 31	mpbid 210	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> F Fn { X } )
33		snidg 4040	. . . . . . . 8 |- ( X e. V -> X e. { X } )
34	33	3ad2ant1 1018	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> X e. { X } )
35		fnsnfv 5918	. . . . . . . 8 |- ( ( F Fn { X } /\ X e. { X } ) -> { ( F ` X ) } = ( F " { X } ) )
36	35	eqcomd 2451	. . . . . . 7 |- ( ( F Fn { X } /\ X e. { X } ) -> ( F " { X } ) = { ( F ` X ) } )
37	32, 34, 36	syl2anc 661	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( F " { X } ) = { ( F ` X ) } )
38	37	neeq1d 2720	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( ( F " { X } ) =/= { Z } <-> { ( F ` X ) } =/= { Z } ) )
39	5	fveq1i 5857	. . . . . . . 8 |- ( F ` X ) = ( { <. X , Y >. } ` X )
40		fvsng 6090	. . . . . . . . 9 |- ( ( X e. V /\ Y e. W ) -> ( { <. X , Y >. } ` X ) = Y )
41	40	3adant3 1017	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { <. X , Y >. } ` X ) = Y )
42	39, 41	syl5eq 2496	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( F ` X ) = Y )
43	42	sneqd 4026	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { ( F ` X ) } = { Y } )
44	43	neeq1d 2720	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { ( F ` X ) } =/= { Z } <-> { Y } =/= { Z } ) )
45		sneqbg 4185	. . . . . . 7 |- ( Y e. W -> ( { Y } = { Z } <-> Y = Z ) )
46	45	3ad2ant2 1019	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { Y } = { Z } <-> Y = Z ) )
47	46	necon3abid 2689	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { Y } =/= { Z } <-> -. Y = Z ) )
48	38, 44, 47	3bitrd 279	. . . 4 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( ( F " { X } ) =/= { Z } <-> -. Y = Z ) )
49	48	ifbid 3948	. . 3 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> if ( ( F " { X } ) =/= { Z } , { X } , (/) ) = if ( -. Y = Z , { X } , (/) ) )
50		ifnot 3971	. . 3 |- if ( -. Y = Z , { X } , (/) ) = if ( Y = Z , (/) , { X } )
51	49, 50	syl6eq 2500	. 2 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> if ( ( F " { X } ) =/= { Z } , { X } , (/) ) = if ( Y = Z , (/) , { X } ) )
52	14, 26, 51	3eqtrd 2488	1 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( F supp Z ) = if ( Y = Z , (/) , { X } ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 974   = wceq 1383   e. wcel 1804   =/= wne 2638   {crab 2797   _Vcvv 3095   (/)c0 3770   ifcif 3926   {csn 4014   <.cop 4020   dom cdm 4989   "cima 4992   Fn wfn 5573   -->wf 5574   -1-1-onto->wf1o 5577   ` cfv 5578  (class class class)co 6281   supp csupp 6903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-supp 6904

This theorem is referenced by: (None)