Theorem suppsnopdc 6484

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 5680	. . . . . . . 8 |- ( ( X e. V /\ Y e. W ) -> { <. X , Y >. } : { X } -1-1-onto-> { Y } )
5		f1of 5637	. . . . . . . 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 1048	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { <. X , Y >. } : { X } --> { Y } )
8		suppsnop.f	. . . . . . 7 |- F = { <. X , Y >. }
9	8	feq1i 5524	. . . . . 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 1278	. . . 4 |- ( ph -> F : { X } --> { Y } )
12		snexg 4319	. . . . 5 |- ( X e. V -> { X } e. _V )
13	1, 12	syl 14	. . . 4 |- ( ph -> { X } e. _V )
14	11, 13	fexd 5942	. . 3 |- ( ph -> F e. _V )
15		suppval 6471	. . 3 |- ( ( F e. _V /\ Z e. U ) -> ( F supp Z ) = { x e. dom F | ( F " { x } ) =/= { Z } } )
16	14, 3, 15	syl2anc 415	. 2 |- ( ph -> ( F supp Z ) = { x e. dom F | ( F " { x } ) =/= { Z } } )
17	10	fdmd 5538	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> dom F = { X } )
18	17	rabeqdv 2815	. . . 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 3719	. . . . . . 7 |- ( x = X -> { x } = { X } )
20	19	imaeq2d 5124	. . . . . 6 |- ( x = X -> ( F " { x } ) = ( F " { X } ) )
21	20	neeq1d 2438	. . . . 5 |- ( x = X -> ( ( F " { x } ) =/= { Z } <-> ( F " { X } ) =/= { Z } ) )
22	21	rabsnif 3777	. . . 4 |- { x e. { X } | ( F " { x } ) =/= { Z } } = if ( ( F " { X } ) =/= { Z } , { X } , (/) )
23	18, 22	eqtrdi 2287	. . 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 1278	. 2 |- ( ph -> { x e. dom F | ( F " { x } ) =/= { Z } } = if ( ( F " { X } ) =/= { Z } , { X } , (/) ) )
25	10	ffnd 5532	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> F Fn { X } )
26		snidg 3737	. . . . . . . . 9 |- ( X e. V -> X e. { X } )
27	26	3ad2ant1 1049	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> X e. { X } )
28		fnsnfv 5759	. . . . . . . . 9 |- ( ( F Fn { X } /\ X e. { X } ) -> { ( F ` X ) } = ( F " { X } ) )
29	28	eqcomd 2244	. . . . . . . 8 |- ( ( F Fn { X } /\ X e. { X } ) -> ( F " { X } ) = { ( F ` X ) } )
30	25, 27, 29	syl2anc 415	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( F " { X } ) = { ( F ` X ) } )
31	30	neeq1d 2438	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( ( F " { X } ) =/= { Z } <-> { ( F ` X ) } =/= { Z } ) )
32	8	fveq1i 5694	. . . . . . . . 9 |- ( F ` X ) = ( { <. X , Y >. } ` X )
33		fvsng 5905	. . . . . . . . . 10 |- ( ( X e. V /\ Y e. W ) -> ( { <. X , Y >. } ` X ) = Y )
34	33	3adant3 1048	. . . . . . . . 9 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { <. X , Y >. } ` X ) = Y )
35	32, 34	eqtrid 2283	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( F ` X ) = Y )
36	35	sneqd 3721	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { ( F ` X ) } = { Y } )
37	36	neeq1d 2438	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { ( F ` X ) } =/= { Z } <-> { Y } =/= { Z } ) )
38		sneqbg 3886	. . . . . . . 8 |- ( Y e. W -> ( { Y } = { Z } <-> Y = Z ) )
39	38	3ad2ant2 1050	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { Y } = { Z } <-> Y = Z ) )
40	39	necon3abid 2459	. . . . . 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 3662	. . . 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 1278	. . 3 |- ( ph -> if ( ( F " { X } ) =/= { Z } , { X } , (/) ) = if ( -. Y = Z , { X } , (/) ) )
44		suppsnopdc.dc	. . . 4 |- ( ph -> DECID Y = Z )
45		ifnotdc 3679	. . . 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 2271	. 2 |- ( ph -> if ( ( F " { X } ) =/= { Z } , { X } , (/) ) = if ( Y = Z , (/) , { X } ) )
48	16, 24, 47	3eqtrd 2275	1 |- ( ph -> ( F supp Z ) = if ( Y = Z , (/) , { X } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   {crab 2532   _Vcvv 2821   (/)c0 3520   ifcif 3638   {csn 3708   <.cop 3711   dom cdm 4772   "cima 4775   Fn wfn 5370   -->wf 5371   -1-1-onto->wf1o 5374   ` cfv 5375  (class class class)co 6079   supp csupp 6469

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-supp 6470

This theorem is referenced by:  snopfsuppdc  7293