Theorem suppsnop 6955

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 5876	. . . . . . 7 |- ( ( X e. V /\ Y e. W ) -> { <. X , Y >. } : { X } -1-1-onto-> { Y } )
2		f1of 5837	. . . . . . 7 |- ( { <. X , Y >. } : { X } -1-1-onto-> { Y } -> { <. X , Y >. } : { X } --> { Y } )
3	1, 2	syl 17	. . . . . 6 |- ( ( X e. V /\ Y e. W ) -> { <. X , Y >. } : { X } --> { Y } )
4	3	3adant3 1034	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { <. X , Y >. } : { X } --> { Y } )
5		suppsnop.f	. . . . . 6 |- F = { <. X , Y >. }
6	5	feq1i 5742	. . . . 5 |- ( F : { X } --> { Y } <-> { <. X , Y >. } : { X } --> { Y } )
7	4, 6	sylibr 217	. . . 4 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> F : { X } --> { Y } )
8		snex 4655	. . . . 5 |- { X } e. _V
9	8	a1i 11	. . . 4 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { X } e. _V )
10		fex 6163	. . . 4 |- ( ( F : { X } --> { Y } /\ { X } e. _V ) -> F e. _V )
11	7, 9, 10	syl2anc 671	. . 3 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> F e. _V )
12		simp3 1016	. . 3 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> Z e. U )
13		suppval 6943	. . 3 |- ( ( F e. _V /\ Z e. U ) -> ( F supp Z ) = { x e. dom F | ( F " { x } ) =/= { Z } } )
14	11, 12, 13	syl2anc 671	. 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 5056	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> dom F = dom { <. X , Y >. } )
17		dmsnopg 5326	. . . . . 6 |- ( Y e. W -> dom { <. X , Y >. } = { X } )
18	17	3ad2ant2 1036	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> dom { <. X , Y >. } = { X } )
19	16, 18	eqtrd 2496	. . . 4 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> dom F = { X } )
20		rabeq 3050	. . . 4 |- ( dom F = { X } -> { x e. dom F | ( F " { x } ) =/= { Z } } = { x e. { X } | ( F " { x } ) =/= { Z } } )
21	19, 20	syl 17	. . 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 3990	. . . . . 6 |- ( x = X -> { x } = { X } )
23	22	imaeq2d 5187	. . . . 5 |- ( x = X -> ( F " { x } ) = ( F " { X } ) )
24	23	neeq1d 2695	. . . 4 |- ( x = X -> ( ( F " { x } ) =/= { Z } <-> ( F " { X } ) =/= { Z } ) )
25	24	rabsnif 4054	. . 3 |- { x e. { X } | ( F " { x } ) =/= { Z } } = if ( ( F " { X } ) =/= { Z } , { X } , (/) )
26	21, 25	syl6eq 2512	. 2 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { x e. dom F | ( F " { x } ) =/= { Z } } = if ( ( F " { X } ) =/= { Z } , { X } , (/) ) )
27		fnsng 5648	. . . . . . . . 9 |- ( ( X e. V /\ Y e. W ) -> { <. X , Y >. } Fn { X } )
28	27	3adant3 1034	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { <. X , Y >. } Fn { X } )
29	5	eqcomi 2471	. . . . . . . . . 10 |- { <. X , Y >. } = F
30	29	a1i 11	. . . . . . . . 9 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { <. X , Y >. } = F )
31	30	fneq1d 5688	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { <. X , Y >. } Fn { X } <-> F Fn { X } ) )
32	28, 31	mpbid 215	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> F Fn { X } )
33		snidg 4006	. . . . . . . 8 |- ( X e. V -> X e. { X } )
34	33	3ad2ant1 1035	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> X e. { X } )
35		fnsnfv 5948	. . . . . . . 8 |- ( ( F Fn { X } /\ X e. { X } ) -> { ( F ` X ) } = ( F " { X } ) )
36	35	eqcomd 2468	. . . . . . 7 |- ( ( F Fn { X } /\ X e. { X } ) -> ( F " { X } ) = { ( F ` X ) } )
37	32, 34, 36	syl2anc 671	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( F " { X } ) = { ( F ` X ) } )
38	37	neeq1d 2695	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( ( F " { X } ) =/= { Z } <-> { ( F ` X ) } =/= { Z } ) )
39	5	fveq1i 5889	. . . . . . . 8 |- ( F ` X ) = ( { <. X , Y >. } ` X )
40		fvsng 6122	. . . . . . . . 9 |- ( ( X e. V /\ Y e. W ) -> ( { <. X , Y >. } ` X ) = Y )
41	40	3adant3 1034	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { <. X , Y >. } ` X ) = Y )
42	39, 41	syl5eq 2508	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( F ` X ) = Y )
43	42	sneqd 3992	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { ( F ` X ) } = { Y } )
44	43	neeq1d 2695	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { ( F ` X ) } =/= { Z } <-> { Y } =/= { Z } ) )
45		sneqbg 4155	. . . . . . 7 |- ( Y e. W -> ( { Y } = { Z } <-> Y = Z ) )
46	45	3ad2ant2 1036	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { Y } = { Z } <-> Y = Z ) )
47	46	necon3abid 2672	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { Y } =/= { Z } <-> -. Y = Z ) )
48	38, 44, 47	3bitrd 287	. . . 4 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( ( F " { X } ) =/= { Z } <-> -. Y = Z ) )
49	48	ifbid 3915	. . 3 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> if ( ( F " { X } ) =/= { Z } , { X } , (/) ) = if ( -. Y = Z , { X } , (/) ) )
50		ifnot 3938	. . 3 |- if ( -. Y = Z , { X } , (/) ) = if ( Y = Z , (/) , { X } )
51	49, 50	syl6eq 2512	. 2 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> if ( ( F " { X } ) =/= { Z } , { X } , (/) ) = if ( Y = Z , (/) , { X } ) )
52	14, 26, 51	3eqtrd 2500	1 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( F supp Z ) = if ( Y = Z , (/) , { X } ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 991   = wceq 1455   e. wcel 1898   =/= wne 2633   {crab 2753   _Vcvv 3057   (/)c0 3743   ifcif 3893   {csn 3980   <.cop 3986   dom cdm 4853   "cima 4856   Fn wfn 5596   -->wf 5597   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6315   supp csupp 6941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pr 4653  ax-un 6610

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-supp 6942

This theorem is referenced by: (None)