Theorem suppsnopdc 6463

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

Hypotheses

Ref	Expression
suppsnop.f	|- F = { <. X , Y >. }
suppsnopdc.x	|- ( ph -> X e. V )
suppsnopdc.y	|- ( ph -> Y e. W )
suppsnopdc.z	|- ( ph -> Z e. U )
suppsnopdc.dc	|- ( ph -> DECID Y = Z )

Assertion

Ref	Expression
suppsnopdc	|- ( ph -> ( F supp Z ) = if ( Y = Z , (/) , { X } ) )

Proof of Theorem suppsnopdc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suppsnopdc.x	. . . . 5 |- ( ph -> X e. V )
2		suppsnopdc.y	. . . . 5 |- ( ph -> Y e. W )
3		suppsnopdc.z	. . . . 5 |- ( ph -> Z e. U )
4		f1osng 5662	. . . . . . . 8 |- ( ( X e. V /\ Y e. W ) -> { <. X , Y >. } : { X } -1-1-onto-> { Y } )
5		f1of 5619	. . . . . . . 8 |- ( { <. X , Y >. } : { X } -1-1-onto-> { Y } -> { <. X , Y >. } : { X } --> { Y } )
6	4, 5	syl 14	. . . . . . 7 |- ( ( X e. V /\ Y e. W ) -> { <. X , Y >. } : { X } --> { Y } )
7	6	3adant3 1044	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { <. X , Y >. } : { X } --> { Y } )
8		suppsnop.f	. . . . . . 7 |- F = { <. X , Y >. }
9	8	feq1i 5506	. . . . . 6 |- ( F : { X } --> { Y } <-> { <. X , Y >. } : { X } --> { Y } )
10	7, 9	sylibr 134	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> F : { X } --> { Y } )
11	1, 2, 3, 10	syl3anc 1274	. . . 4 |- ( ph -> F : { X } --> { Y } )
12		snexg 4302	. . . . 5 |- ( X e. V -> { X } e. _V )
13	1, 12	syl 14	. . . 4 |- ( ph -> { X } e. _V )
14	11, 13	fexd 5921	. . 3 |- ( ph -> F e. _V )
15		suppval 6450	. . 3 |- ( ( F e. _V /\ Z e. U ) -> ( F supp Z ) = { x e. dom F | ( F " { x } ) =/= { Z } } )
16	14, 3, 15	syl2anc 411	. 2 |- ( ph -> ( F supp Z ) = { x e. dom F | ( F " { x } ) =/= { Z } } )
17	10	fdmd 5520	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> dom F = { X } )
18	17	rabeqdv 2809	. . . 4 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { x e. dom F | ( F " { x } ) =/= { Z } } = { x e. { X } | ( F " { x } ) =/= { Z } } )
19		sneq 3705	. . . . . . 7 |- ( x = X -> { x } = { X } )
20	19	imaeq2d 5106	. . . . . 6 |- ( x = X -> ( F " { x } ) = ( F " { X } ) )
21	20	neeq1d 2432	. . . . 5 |- ( x = X -> ( ( F " { x } ) =/= { Z } <-> ( F " { X } ) =/= { Z } ) )
22	21	rabsnif 3763	. . . 4 |- { x e. { X } | ( F " { x } ) =/= { Z } } = if ( ( F " { X } ) =/= { Z } , { X } , (/) )
23	18, 22	eqtrdi 2283	. . 3 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { x e. dom F | ( F " { x } ) =/= { Z } } = if ( ( F " { X } ) =/= { Z } , { X } , (/) ) )
24	1, 2, 3, 23	syl3anc 1274	. 2 |- ( ph -> { x e. dom F | ( F " { x } ) =/= { Z } } = if ( ( F " { X } ) =/= { Z } , { X } , (/) ) )
25	10	ffnd 5514	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> F Fn { X } )
26		snidg 3723	. . . . . . . . 9 |- ( X e. V -> X e. { X } )
27	26	3ad2ant1 1045	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> X e. { X } )
28		fnsnfv 5741	. . . . . . . . 9 |- ( ( F Fn { X } /\ X e. { X } ) -> { ( F ` X ) } = ( F " { X } ) )
29	28	eqcomd 2240	. . . . . . . 8 |- ( ( F Fn { X } /\ X e. { X } ) -> ( F " { X } ) = { ( F ` X ) } )
30	25, 27, 29	syl2anc 411	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( F " { X } ) = { ( F ` X ) } )
31	30	neeq1d 2432	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( ( F " { X } ) =/= { Z } <-> { ( F ` X ) } =/= { Z } ) )
32	8	fveq1i 5676	. . . . . . . . 9 |- ( F ` X ) = ( { <. X , Y >. } ` X )
33		fvsng 5885	. . . . . . . . . 10 |- ( ( X e. V /\ Y e. W ) -> ( { <. X , Y >. } ` X ) = Y )
34	33	3adant3 1044	. . . . . . . . 9 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { <. X , Y >. } ` X ) = Y )
35	32, 34	eqtrid 2279	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( F ` X ) = Y )
36	35	sneqd 3707	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { ( F ` X ) } = { Y } )
37	36	neeq1d 2432	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { ( F ` X ) } =/= { Z } <-> { Y } =/= { Z } ) )
38		sneqbg 3872	. . . . . . . 8 |- ( Y e. W -> ( { Y } = { Z } <-> Y = Z ) )
39	38	3ad2ant2 1046	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { Y } = { Z } <-> Y = Z ) )
40	39	necon3abid 2453	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { Y } =/= { Z } <-> -. Y = Z ) )
41	31, 37, 40	3bitrd 214	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( ( F " { X } ) =/= { Z } <-> -. Y = Z ) )
42	41	ifbid 3648	. . . 4 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> if ( ( F " { X } ) =/= { Z } , { X } , (/) ) = if ( -. Y = Z , { X } , (/) ) )
43	1, 2, 3, 42	syl3anc 1274	. . 3 |- ( ph -> if ( ( F " { X } ) =/= { Z } , { X } , (/) ) = if ( -. Y = Z , { X } , (/) ) )
44		suppsnopdc.dc	. . . 4 |- ( ph -> DECID Y = Z )
45		ifnotdc 3665	. . . 4 |- (DECID Y = Z -> if ( -. Y = Z , { X } , (/) ) = if ( Y = Z , (/) , { X } ) )
46	44, 45	syl 14	. . 3 |- ( ph -> if ( -. Y = Z , { X } , (/) ) = if ( Y = Z , (/) , { X } ) )
47	43, 46	eqtrd 2267	. 2 |- ( ph -> if ( ( F " { X } ) =/= { Z } , { X } , (/) ) = if ( Y = Z , (/) , { X } ) )
48	16, 24, 47	3eqtrd 2271	1 |- ( ph -> ( F supp Z ) = if ( Y = Z , (/) , { X } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   {crab 2526   _Vcvv 2815   (/)c0 3512   ifcif 3624   {csn 3694   <.cop 3697   dom cdm 4754   "cima 4757   Fn wfn 5352   -->wf 5353   -1-1-onto->wf1o 5356   ` cfv 5357  (class class class)co 6058   supp csupp 6448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-supp 6449

This theorem is referenced by:  snopfsuppdc  7265