Theorem suppsnopdc 6449

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 5656	. . . . . . . 8 |- ( ( X e. V /\ Y e. W ) -> { <. X , Y >. } : { X } -1-1-onto-> { Y } )
5		f1of 5613	. . . . . . . 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 5500	. . . . . 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 4296	. . . . 5 |- ( X e. V -> { X } e. _V )
13	1, 12	syl 14	. . . 4 |- ( ph -> { X } e. _V )
14	11, 13	fexd 5915	. . 3 |- ( ph -> F e. _V )
15		suppval 6436	. . 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 5514	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> dom F = { X } )
18	17	rabeqdv 2806	. . . 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 3699	. . . . . . 7 |- ( x = X -> { x } = { X } )
20	19	imaeq2d 5100	. . . . . 6 |- ( x = X -> ( F " { x } ) = ( F " { X } ) )
21	20	neeq1d 2430	. . . . 5 |- ( x = X -> ( ( F " { x } ) =/= { Z } <-> ( F " { X } ) =/= { Z } ) )
22	21	rabsnif 3757	. . . 4 |- { x e. { X } | ( F " { x } ) =/= { Z } } = if ( ( F " { X } ) =/= { Z } , { X } , (/) )
23	18, 22	eqtrdi 2281	. . 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 5508	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> F Fn { X } )
26		snidg 3717	. . . . . . . . 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 5735	. . . . . . . . 9 |- ( ( F Fn { X } /\ X e. { X } ) -> { ( F ` X ) } = ( F " { X } ) )
29	28	eqcomd 2238	. . . . . . . 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 2430	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( ( F " { X } ) =/= { Z } <-> { ( F ` X ) } =/= { Z } ) )
32	8	fveq1i 5670	. . . . . . . . 9 |- ( F ` X ) = ( { <. X , Y >. } ` X )
33		fvsng 5879	. . . . . . . . . 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 2277	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( F ` X ) = Y )
36	35	sneqd 3701	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { ( F ` X ) } = { Y } )
37	36	neeq1d 2430	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { ( F ` X ) } =/= { Z } <-> { Y } =/= { Z } ) )
38		sneqbg 3866	. . . . . . . 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 2451	. . . . . 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 3643	. . . 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 3660	. . . 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 2265	. 2 |- ( ph -> if ( ( F " { X } ) =/= { Z } , { X } , (/) ) = if ( Y = Z , (/) , { X } ) )
48	16, 24, 47	3eqtrd 2269	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 2203   =/= wne 2412   {crab 2524   _Vcvv 2812   (/)c0 3507   ifcif 3619   {csn 3688   <.cop 3691   dom cdm 4748   "cima 4751   Fn wfn 5346   -->wf 5347   -1-1-onto->wf1o 5350   ` cfv 5351  (class class class)co 6049   supp csupp 6434

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-supp 6435

This theorem is referenced by:  snopfsuppdc  7251