Theorem tfr0dm 5991

Description: Transfinite recursion is defined at the empty set. (Contributed by Jim Kingdon, 8-Mar-2022.)

Hypothesis

Ref	Expression
tfr.1	|- F = recs ( G )

Assertion

Ref	Expression
tfr0dm	|- ( ( G ` (/) ) e. V -> (/) e. dom F )

Proof of Theorem tfr0dm

Dummy variables x f y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 3925	. . . . 5 |- (/) e. _V
2		opexg 4011	. . . . 5 |- ( ( (/) e. _V /\ ( G ` (/) ) e. V ) -> <. (/) , ( G ` (/) ) >. e. _V )
3	1, 2	mpan 415	. . . 4 |- ( ( G ` (/) ) e. V -> <. (/) , ( G ` (/) ) >. e. _V )
4		snidg 3441	. . . 4 |- ( <. (/) , ( G ` (/) ) >. e. _V -> <. (/) , ( G ` (/) ) >. e. { <. (/) , ( G ` (/) ) >. } )
5	3, 4	syl 14	. . 3 |- ( ( G ` (/) ) e. V -> <. (/) , ( G ` (/) ) >. e. { <. (/) , ( G ` (/) ) >. } )
6		fnsng 4997	. . . . 5 |- ( ( (/) e. _V /\ ( G ` (/) ) e. V ) -> { <. (/) , ( G ` (/) ) >. } Fn { (/) } )
7	1, 6	mpan 415	. . . 4 |- ( ( G ` (/) ) e. V -> { <. (/) , ( G ` (/) ) >. } Fn { (/) } )
8		fvsng 5411	. . . . . . 7 |- ( ( (/) e. _V /\ ( G ` (/) ) e. V ) -> ( { <. (/) , ( G ` (/) ) >. } ` (/) ) = ( G ` (/) ) )
9	1, 8	mpan 415	. . . . . 6 |- ( ( G ` (/) ) e. V -> ( { <. (/) , ( G ` (/) ) >. } ` (/) ) = ( G ` (/) ) )
10		res0 4664	. . . . . . 7 |- ( { <. (/) , ( G ` (/) ) >. } |` (/) ) = (/)
11	10	fveq2i 5232	. . . . . 6 |- ( G ` ( { <. (/) , ( G ` (/) ) >. } |` (/) ) ) = ( G ` (/) )
12	9, 11	syl6eqr 2133	. . . . 5 |- ( ( G ` (/) ) e. V -> ( { <. (/) , ( G ` (/) ) >. } ` (/) ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` (/) ) ) )
13		fveq2 5229	. . . . . . 7 |- ( y = (/) -> ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( { <. (/) , ( G ` (/) ) >. } ` (/) ) )
14		reseq2 4655	. . . . . . . 8 |- ( y = (/) -> ( { <. (/) , ( G ` (/) ) >. } |` y ) = ( { <. (/) , ( G ` (/) ) >. } |` (/) ) )
15	14	fveq2d 5233	. . . . . . 7 |- ( y = (/) -> ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` (/) ) ) )
16	13, 15	eqeq12d 2097	. . . . . 6 |- ( y = (/) -> ( ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) <-> ( { <. (/) , ( G ` (/) ) >. } ` (/) ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` (/) ) ) ) )
17	1, 16	ralsn 3454	. . . . 5 |- ( A. y e. { (/) } ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) <-> ( { <. (/) , ( G ` (/) ) >. } ` (/) ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` (/) ) ) )
18	12, 17	sylibr 132	. . . 4 |- ( ( G ` (/) ) e. V -> A. y e. { (/) } ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) )
19		suc0 4194	. . . . . 6 |- suc (/) = { (/) }
20		0elon 4175	. . . . . . 7 |- (/) e. On
21	20	onsuci 4288	. . . . . 6 |- suc (/) e. On
22	19, 21	eqeltrri 2156	. . . . 5 |- { (/) } e. On
23		fneq2 5039	. . . . . . 7 |- ( x = { (/) } -> ( { <. (/) , ( G ` (/) ) >. } Fn x <-> { <. (/) , ( G ` (/) ) >. } Fn { (/) } ) )
24		raleq 2554	. . . . . . 7 |- ( x = { (/) } -> ( A. y e. x ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) <-> A. y e. { (/) } ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) ) )
25	23, 24	anbi12d 457	. . . . . 6 |- ( x = { (/) } -> ( ( { <. (/) , ( G ` (/) ) >. } Fn x /\ A. y e. x ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) ) <-> ( { <. (/) , ( G ` (/) ) >. } Fn { (/) } /\ A. y e. { (/) } ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) ) ) )
26	25	rspcev 2710	. . . . 5 |- ( ( { (/) } e. On /\ ( { <. (/) , ( G ` (/) ) >. } Fn { (/) } /\ A. y e. { (/) } ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) ) ) -> E. x e. On ( { <. (/) , ( G ` (/) ) >. } Fn x /\ A. y e. x ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) ) )
27	22, 26	mpan 415	. . . 4 |- ( ( { <. (/) , ( G ` (/) ) >. } Fn { (/) } /\ A. y e. { (/) } ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) ) -> E. x e. On ( { <. (/) , ( G ` (/) ) >. } Fn x /\ A. y e. x ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) ) )
28	7, 18, 27	syl2anc 403	. . 3 |- ( ( G ` (/) ) e. V -> E. x e. On ( { <. (/) , ( G ` (/) ) >. } Fn x /\ A. y e. x ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) ) )
29		snexg 3976	. . . . 5 |- ( <. (/) , ( G ` (/) ) >. e. _V -> { <. (/) , ( G ` (/) ) >. } e. _V )
30		eleq2 2146	. . . . . . 7 |- ( f = { <. (/) , ( G ` (/) ) >. } -> ( <. (/) , ( G ` (/) ) >. e. f <-> <. (/) , ( G ` (/) ) >. e. { <. (/) , ( G ` (/) ) >. } ) )
31		fneq1 5038	. . . . . . . . 9 |- ( f = { <. (/) , ( G ` (/) ) >. } -> ( f Fn x <-> { <. (/) , ( G ` (/) ) >. } Fn x ) )
32		fveq1 5228	. . . . . . . . . . 11 |- ( f = { <. (/) , ( G ` (/) ) >. } -> ( f ` y ) = ( { <. (/) , ( G ` (/) ) >. } ` y ) )
33		reseq1 4654	. . . . . . . . . . . 12 |- ( f = { <. (/) , ( G ` (/) ) >. } -> ( f |` y ) = ( { <. (/) , ( G ` (/) ) >. } |` y ) )
34	33	fveq2d 5233	. . . . . . . . . . 11 |- ( f = { <. (/) , ( G ` (/) ) >. } -> ( G ` ( f |` y ) ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) )
35	32, 34	eqeq12d 2097	. . . . . . . . . 10 |- ( f = { <. (/) , ( G ` (/) ) >. } -> ( ( f ` y ) = ( G ` ( f |` y ) ) <-> ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) ) )
36	35	ralbidv 2373	. . . . . . . . 9 |- ( f = { <. (/) , ( G ` (/) ) >. } -> ( A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) <-> A. y e. x ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) ) )
37	31, 36	anbi12d 457	. . . . . . . 8 |- ( f = { <. (/) , ( G ` (/) ) >. } -> ( ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) <-> ( { <. (/) , ( G ` (/) ) >. } Fn x /\ A. y e. x ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) ) ) )
38	37	rexbidv 2374	. . . . . . 7 |- ( f = { <. (/) , ( G ` (/) ) >. } -> ( E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) <-> E. x e. On ( { <. (/) , ( G ` (/) ) >. } Fn x /\ A. y e. x ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) ) ) )
39	30, 38	anbi12d 457	. . . . . 6 |- ( f = { <. (/) , ( G ` (/) ) >. } -> ( ( <. (/) , ( G ` (/) ) >. e. f /\ E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) ) <-> ( <. (/) , ( G ` (/) ) >. e. { <. (/) , ( G ` (/) ) >. } /\ E. x e. On ( { <. (/) , ( G ` (/) ) >. } Fn x /\ A. y e. x ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) ) ) ) )
40	39	spcegv 2695	. . . . 5 |- ( { <. (/) , ( G ` (/) ) >. } e. _V -> ( ( <. (/) , ( G ` (/) ) >. e. { <. (/) , ( G ` (/) ) >. } /\ E. x e. On ( { <. (/) , ( G ` (/) ) >. } Fn x /\ A. y e. x ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) ) ) -> E. f ( <. (/) , ( G ` (/) ) >. e. f /\ E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) ) ) )
41	3, 29, 40	3syl 17	. . . 4 |- ( ( G ` (/) ) e. V -> ( ( <. (/) , ( G ` (/) ) >. e. { <. (/) , ( G ` (/) ) >. } /\ E. x e. On ( { <. (/) , ( G ` (/) ) >. } Fn x /\ A. y e. x ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) ) ) -> E. f ( <. (/) , ( G ` (/) ) >. e. f /\ E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) ) ) )
42		tfr.1	. . . . . 6 |- F = recs ( G )
43	42	eleq2i 2149	. . . . 5 |- ( <. (/) , ( G ` (/) ) >. e. F <-> <. (/) , ( G ` (/) ) >. e. recs ( G ) )
44		df-recs 5974	. . . . . 6 |- recs ( G ) = U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
45	44	eleq2i 2149	. . . . 5 |- ( <. (/) , ( G ` (/) ) >. e. recs ( G ) <-> <. (/) , ( G ` (/) ) >. e. U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } )
46		eluniab 3633	. . . . 5 |- ( <. (/) , ( G ` (/) ) >. e. U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } <-> E. f ( <. (/) , ( G ` (/) ) >. e. f /\ E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) ) )
47	43, 45, 46	3bitri 204	. . . 4 |- ( <. (/) , ( G ` (/) ) >. e. F <-> E. f ( <. (/) , ( G ` (/) ) >. e. f /\ E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) ) )
48	41, 47	syl6ibr 160	. . 3 |- ( ( G ` (/) ) e. V -> ( ( <. (/) , ( G ` (/) ) >. e. { <. (/) , ( G ` (/) ) >. } /\ E. x e. On ( { <. (/) , ( G ` (/) ) >. } Fn x /\ A. y e. x ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) ) ) -> <. (/) , ( G ` (/) ) >. e. F ) )
49	5, 28, 48	mp2and 424	. 2 |- ( ( G ` (/) ) e. V -> <. (/) , ( G ` (/) ) >. e. F )
50		opeldmg 4588	. . 3 |- ( ( (/) e. _V /\ ( G ` (/) ) e. V ) -> ( <. (/) , ( G ` (/) ) >. e. F -> (/) e. dom F ) )
51	1, 50	mpan 415	. 2 |- ( ( G ` (/) ) e. V -> ( <. (/) , ( G ` (/) ) >. e. F -> (/) e. dom F ) )
52	49, 51	mpd 13	1 |- ( ( G ` (/) ) e. V -> (/) e. dom F )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285   E.wex 1422   e. wcel 1434   {cab 2069   A.wral 2353   E.wrex 2354   _Vcvv 2610   (/)c0 3267   {csn 3416   <.cop 3419   U.cuni 3621   Oncon0 4146   suc csuc 4148   dom cdm 4391   |` cres 4393   Fn wfn 4947   ` cfv 4952  recscrecs 5973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-res 4403  df-iota 4917  df-fun 4954  df-fn 4955  df-fv 4960  df-recs 5974

This theorem is referenced by:  tfr0  5992