Theorem suppsnop 6817

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 5790	. . . . . . 7 |- ( ( X e. V /\ Y e. W ) -> { <. X , Y >. } : { X } -1-1-onto-> { Y } )
2		f1of 5752	. . . . . . 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 1008	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { <. X , Y >. } : { X } --> { Y } )
5		suppsnop.f	. . . . . 6 |- F = { <. X , Y >. }
6	5	feq1i 5662	. . . . 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 4644	. . . . 5 |- { X } e. _V
9	8	a1i 11	. . . 4 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { X } e. _V )
10		fex 6062	. . . 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 990	. . 3 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> Z e. U )
13		suppval 6805	. . 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	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> F = { <. X , Y >. } )
16	15	dmeqd 5153	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> dom F = dom { <. X , Y >. } )
17		dmsnopg 5421	. . . . . . 7 |- ( Y e. W -> dom { <. X , Y >. } = { X } )
18	17	3ad2ant2 1010	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> dom { <. X , Y >. } = { X } )
19	16, 18	eqtrd 2495	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> dom F = { X } )
20		rabeq 3072	. . . . 5 |- ( dom F = { X } -> { x e. dom F | ( F " { x } ) =/= { Z } } = { x e. { X } | ( F " { x } ) =/= { Z } } )
21	19, 20	syl 16	. . . 4 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { x e. dom F | ( F " { x } ) =/= { Z } } = { x e. { X } | ( F " { x } ) =/= { Z } } )
22		sneq 3998	. . . . . . 7 |- ( x = X -> { x } = { X } )
23	22	imaeq2d 5280	. . . . . 6 |- ( x = X -> ( F " { x } ) = ( F " { X } ) )
24	23	neeq1d 2729	. . . . 5 |- ( x = X -> ( ( F " { x } ) =/= { Z } <-> ( F " { X } ) =/= { Z } ) )
25	24	rabsnif 4055	. . . 4 |- { x e. { X } | ( F " { x } ) =/= { Z } } = if ( ( F " { X } ) =/= { Z } , { X } , (/) )
26	21, 25	syl6eq 2511	. . 3 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { x e. dom F | ( F " { x } ) =/= { Z } } = if ( ( F " { X } ) =/= { Z } , { X } , (/) ) )
27		fnsng 5576	. . . . . . . . . 10 |- ( ( X e. V /\ Y e. W ) -> { <. X , Y >. } Fn { X } )
28	27	3adant3 1008	. . . . . . . . 9 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { <. X , Y >. } Fn { X } )
29	5	eqcomi 2467	. . . . . . . . . . 11 |- { <. X , Y >. } = F
30	29	a1i 11	. . . . . . . . . 10 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { <. X , Y >. } = F )
31	30	fneq1d 5612	. . . . . . . . 9 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { <. X , Y >. } Fn { X } <-> F Fn { X } ) )
32	28, 31	mpbid 210	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> F Fn { X } )
33		snidg 4014	. . . . . . . . 9 |- ( X e. V -> X e. { X } )
34	33	3ad2ant1 1009	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> X e. { X } )
35		fnsnfv 5863	. . . . . . . . 9 |- ( ( F Fn { X } /\ X e. { X } ) -> { ( F ` X ) } = ( F " { X } ) )
36	35	eqcomd 2462	. . . . . . . 8 |- ( ( F Fn { X } /\ X e. { X } ) -> ( F " { X } ) = { ( F ` X ) } )
37	32, 34, 36	syl2anc 661	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( F " { X } ) = { ( F ` X ) } )
38	37	neeq1d 2729	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( ( F " { X } ) =/= { Z } <-> { ( F ` X ) } =/= { Z } ) )
39	5	fveq1i 5803	. . . . . . . . 9 |- ( F ` X ) = ( { <. X , Y >. } ` X )
40		fvsng 6024	. . . . . . . . . 10 |- ( ( X e. V /\ Y e. W ) -> ( { <. X , Y >. } ` X ) = Y )
41	40	3adant3 1008	. . . . . . . . 9 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { <. X , Y >. } ` X ) = Y )
42	39, 41	syl5eq 2507	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( F ` X ) = Y )
43	42	sneqd 4000	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { ( F ` X ) } = { Y } )
44	43	neeq1d 2729	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { ( F ` X ) } =/= { Z } <-> { Y } =/= { Z } ) )
45		sneqbg 4154	. . . . . . . 8 |- ( Y e. W -> ( { Y } = { Z } <-> Y = Z ) )
46	45	3ad2ant2 1010	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { Y } = { Z } <-> Y = Z ) )
47	46	necon3abid 2698	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { Y } =/= { Z } <-> -. Y = Z ) )
48	38, 44, 47	3bitrd 279	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( ( F " { X } ) =/= { Z } <-> -. Y = Z ) )
49	48	ifbid 3922	. . . 4 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> if ( ( F " { X } ) =/= { Z } , { X } , (/) ) = if ( -. Y = Z , { X } , (/) ) )
50		ifnot 3945	. . . 4 |- if ( -. Y = Z , { X } , (/) ) = if ( Y = Z , (/) , { X } )
51	49, 50	syl6eq 2511	. . 3 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> if ( ( F " { X } ) =/= { Z } , { X } , (/) ) = if ( Y = Z , (/) , { X } ) )
52	26, 51	eqtrd 2495	. 2 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { x e. dom F | ( F " { x } ) =/= { Z } } = if ( Y = Z , (/) , { X } ) )
53	14, 52	eqtrd 2495	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 965   = wceq 1370   e. wcel 1758   =/= wne 2648   {crab 2803   _Vcvv 3078   (/)c0 3748   ifcif 3902   {csn 3988   <.cop 3994   dom cdm 4951   "cima 4954   Fn wfn 5524   -->wf 5525   -1-1-onto->wf1o 5528   ` cfv 5529  (class class class)co 6203   supp csupp 6803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-supp 6804

This theorem is referenced by: (None)