Theorem suppsnop 6831

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 5762	. . . . . . 7 |- ( ( X e. V /\ Y e. W ) -> { <. X , Y >. } : { X } -1-1-onto-> { Y } )
2		f1of 5724	. . . . . . 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 1014	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { <. X , Y >. } : { X } --> { Y } )
5		suppsnop.f	. . . . . 6 |- F = { <. X , Y >. }
6	5	feq1i 5631	. . . . 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 4603	. . . . 5 |- { X } e. _V
9	8	a1i 11	. . . 4 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { X } e. _V )
10		fex 6046	. . . 4 |- ( ( F : { X } --> { Y } /\ { X } e. _V ) -> F e. _V )
11	7, 9, 10	syl2anc 659	. . 3 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> F e. _V )
12		simp3 996	. . 3 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> Z e. U )
13		suppval 6819	. . 3 |- ( ( F e. _V /\ Z e. U ) -> ( F supp Z ) = { x e. dom F | ( F " { x } ) =/= { Z } } )
14	11, 12, 13	syl2anc 659	. 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 5118	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> dom F = dom { <. X , Y >. } )
17		dmsnopg 5387	. . . . . 6 |- ( Y e. W -> dom { <. X , Y >. } = { X } )
18	17	3ad2ant2 1016	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> dom { <. X , Y >. } = { X } )
19	16, 18	eqtrd 2423	. . . 4 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> dom F = { X } )
20		rabeq 3028	. . . 4 |- ( dom F = { X } -> { x e. dom F | ( F " { x } ) =/= { Z } } = { x e. { X } | ( F " { x } ) =/= { Z } } )
21	19, 20	syl 16	. . 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 3954	. . . . . 6 |- ( x = X -> { x } = { X } )
23	22	imaeq2d 5249	. . . . 5 |- ( x = X -> ( F " { x } ) = ( F " { X } ) )
24	23	neeq1d 2659	. . . 4 |- ( x = X -> ( ( F " { x } ) =/= { Z } <-> ( F " { X } ) =/= { Z } ) )
25	24	rabsnif 4013	. . 3 |- { x e. { X } | ( F " { x } ) =/= { Z } } = if ( ( F " { X } ) =/= { Z } , { X } , (/) )
26	21, 25	syl6eq 2439	. 2 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { x e. dom F | ( F " { x } ) =/= { Z } } = if ( ( F " { X } ) =/= { Z } , { X } , (/) ) )
27		fnsng 5543	. . . . . . . . 9 |- ( ( X e. V /\ Y e. W ) -> { <. X , Y >. } Fn { X } )
28	27	3adant3 1014	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { <. X , Y >. } Fn { X } )
29	5	eqcomi 2395	. . . . . . . . . 10 |- { <. X , Y >. } = F
30	29	a1i 11	. . . . . . . . 9 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { <. X , Y >. } = F )
31	30	fneq1d 5579	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { <. X , Y >. } Fn { X } <-> F Fn { X } ) )
32	28, 31	mpbid 210	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> F Fn { X } )
33		snidg 3970	. . . . . . . 8 |- ( X e. V -> X e. { X } )
34	33	3ad2ant1 1015	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> X e. { X } )
35		fnsnfv 5834	. . . . . . . 8 |- ( ( F Fn { X } /\ X e. { X } ) -> { ( F ` X ) } = ( F " { X } ) )
36	35	eqcomd 2390	. . . . . . 7 |- ( ( F Fn { X } /\ X e. { X } ) -> ( F " { X } ) = { ( F ` X ) } )
37	32, 34, 36	syl2anc 659	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( F " { X } ) = { ( F ` X ) } )
38	37	neeq1d 2659	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( ( F " { X } ) =/= { Z } <-> { ( F ` X ) } =/= { Z } ) )
39	5	fveq1i 5775	. . . . . . . 8 |- ( F ` X ) = ( { <. X , Y >. } ` X )
40		fvsng 6007	. . . . . . . . 9 |- ( ( X e. V /\ Y e. W ) -> ( { <. X , Y >. } ` X ) = Y )
41	40	3adant3 1014	. . . . . . . 8 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { <. X , Y >. } ` X ) = Y )
42	39, 41	syl5eq 2435	. . . . . . 7 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( F ` X ) = Y )
43	42	sneqd 3956	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> { ( F ` X ) } = { Y } )
44	43	neeq1d 2659	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { ( F ` X ) } =/= { Z } <-> { Y } =/= { Z } ) )
45		sneqbg 4114	. . . . . . 7 |- ( Y e. W -> ( { Y } = { Z } <-> Y = Z ) )
46	45	3ad2ant2 1016	. . . . . 6 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { Y } = { Z } <-> Y = Z ) )
47	46	necon3abid 2628	. . . . 5 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( { Y } =/= { Z } <-> -. Y = Z ) )
48	38, 44, 47	3bitrd 279	. . . 4 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> ( ( F " { X } ) =/= { Z } <-> -. Y = Z ) )
49	48	ifbid 3879	. . 3 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> if ( ( F " { X } ) =/= { Z } , { X } , (/) ) = if ( -. Y = Z , { X } , (/) ) )
50		ifnot 3902	. . 3 |- if ( -. Y = Z , { X } , (/) ) = if ( Y = Z , (/) , { X } )
51	49, 50	syl6eq 2439	. 2 |- ( ( X e. V /\ Y e. W /\ Z e. U ) -> if ( ( F " { X } ) =/= { Z } , { X } , (/) ) = if ( Y = Z , (/) , { X } ) )
52	14, 26, 51	3eqtrd 2427	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 367   /\ w3a 971   = wceq 1399   e. wcel 1826   =/= wne 2577   {crab 2736   _Vcvv 3034   (/)c0 3711   ifcif 3857   {csn 3944   <.cop 3950   dom cdm 4913   "cima 4916   Fn wfn 5491   -->wf 5492   -1-1-onto->wf1o 5495   ` cfv 5496  (class class class)co 6196   supp csupp 6817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pr 4601  ax-un 6491

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-supp 6818

This theorem is referenced by: (None)