Theorem suppsnop 7309

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 6177	. . . . . . 7 |- ( ( X e. V /\ Y e. W ) -> { <. X , Y >. } : { X } -1-1-onto-> { Y } )
2		f1of 6137	. . . . . . 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 1081	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { <. X , Y >. } : { X } --> { Y } )
5		suppsnop.f	. . . . . 6 |- F = { <. X , Y >. }
6	5	feq1i 6036	. . . . 5 |- ( F : { X } --> { Y } <-> { <. X , Y >. } : { X } --> { Y } )
7	4, 6	sylibr 224	. . . 4 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> F : { X } --> { Y } )
8		snex 4908	. . . . 5 |- { X } e. _V
9	8	a1i 11	. . . 4 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { X } e. _V )
10		fex 6490	. . . 4 |- ( ( F : { X } --> { Y } /\ { X } e. _V ) -> F e. _V )
11	7, 9, 10	syl2anc 693	. . 3 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> F e. _V )
12		simp3 1063	. . 3 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> Z e. U )
13		suppval 7297	. . 3 |- ( ( F e. _V /\ Z e. U ) -> ( F supp Z ) = { x e. dom F | ( F " { x } ) =/= { Z } } )
14	11, 12, 13	syl2anc 693	. 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 5326	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> dom F = dom { <. X , Y >. } )
17		dmsnopg 5606	. . . . . 6 |- ( Y e. W -> dom { <. X , Y >. } = { X } )
18	17	3ad2ant2 1083	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> dom { <. X , Y >. } = { X } )
19	16, 18	eqtrd 2656	. . . 4 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> dom F = { X } )
20		rabeq 3192	. . . 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 4187	. . . . . 6 |- ( x = X -> { x } = { X } )
23	22	imaeq2d 5466	. . . . 5 |- ( x = X -> ( F " { x } ) = ( F " { X } ) )
24	23	neeq1d 2853	. . . 4 |- ( x = X -> ( ( F " { x } ) =/= { Z } <-> ( F " { X } ) =/= { Z } ) )
25	24	rabsnif 4258	. . 3 |- { x e. { X } | ( F " { x } ) =/= { Z } } = if ( ( F " { X } ) =/= { Z } , { X } , (/) )
26	21, 25	syl6eq 2672	. 2 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { x e. dom F | ( F " { x } ) =/= { Z } } = if ( ( F " { X } ) =/= { Z } , { X } , (/) ) )
27		fnsng 5938	. . . . . . . . 9 |- ( ( X e. V /\ Y e. W ) -> { <. X , Y >. } Fn { X } )
28	27	3adant3 1081	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { <. X , Y >. } Fn { X } )
29	5	eqcomi 2631	. . . . . . . . . 10 |- { <. X , Y >. } = F
30	29	a1i 11	. . . . . . . . 9 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { <. X , Y >. } = F )
31	30	fneq1d 5981	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { <. X , Y >. } Fn { X } <-> F Fn { X } ) )
32	28, 31	mpbid 222	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> F Fn { X } )
33		snidg 4206	. . . . . . . 8 |- ( X e. V -> X e. { X } )
34	33	3ad2ant1 1082	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> X e. { X } )
35		fnsnfv 6258	. . . . . . . 8 |- ( ( F Fn { X } /\ X e. { X } ) -> { ( F ` X ) } = ( F " { X } ) )
36	35	eqcomd 2628	. . . . . . 7 |- ( ( F Fn { X } /\ X e. { X } ) -> ( F " { X } ) = { ( F ` X ) } )
37	32, 34, 36	syl2anc 693	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( F " { X } ) = { ( F ` X ) } )
38	37	neeq1d 2853	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( ( F " { X } ) =/= { Z } <-> { ( F ` X ) } =/= { Z } ) )
39	5	fveq1i 6192	. . . . . . . 8 |- ( F ` X ) = ( { <. X , Y >. } ` X )
40		fvsng 6447	. . . . . . . . 9 |- ( ( X e. V /\ Y e. W ) -> ( { <. X , Y >. } ` X ) = Y )
41	40	3adant3 1081	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { <. X , Y >. } ` X ) = Y )
42	39, 41	syl5eq 2668	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( F ` X ) = Y )
43	42	sneqd 4189	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { ( F ` X ) } = { Y } )
44	43	neeq1d 2853	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { ( F ` X ) } =/= { Z } <-> { Y } =/= { Z } ) )
45		sneqbg 4374	. . . . . . 7 |- ( Y e. W -> ( { Y } = { Z } <-> Y = Z ) )
46	45	3ad2ant2 1083	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { Y } = { Z } <-> Y = Z ) )
47	46	necon3abid 2830	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { Y } =/= { Z } <-> -. Y = Z ) )
48	38, 44, 47	3bitrd 294	. . . 4 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( ( F " { X } ) =/= { Z } <-> -. Y = Z ) )
49	48	ifbid 4108	. . 3 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> if ( ( F " { X } ) =/= { Z } , { X } , (/) ) = if ( -. Y = Z , { X } , (/) ) )
50		ifnot 4133	. . 3 |- if ( -. Y = Z , { X } , (/) ) = if ( Y = Z , (/) , { X } )
51	49, 50	syl6eq 2672	. 2 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> if ( ( F " { X } ) =/= { Z } , { X } , (/) ) = if ( Y = Z , (/) , { X } ) )
52	14, 26, 51	3eqtrd 2660	1 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( F supp Z ) = if ( Y = Z , (/) , { X } ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   (/)c0 3915   ifcif 4086   {csn 4177   <.cop 4183   dom cdm 5114   "cima 5117   Fn wfn 5883   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   supp csupp 7295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-supp 7296

This theorem is referenced by: (None)