Theorem tfr0dm 6326

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 4132	. . . . 5 |- (/) e. _V
2		opexg 4230	. . . . 5 |- ( ( (/) e. _V /\ ( G ` (/) ) e. V ) -> <. (/) , ( G ` (/) ) >. e. _V )
3	1, 2	mpan 424	. . . 4 |- ( ( G ` (/) ) e. V -> <. (/) , ( G ` (/) ) >. e. _V )
4		snidg 3623	. . . 4 |- ( <. (/) , ( G ` (/) ) >. e. _V -> <. (/) , ( G ` (/) ) >. e. { <. (/) , ( G ` (/) ) >. } )
5	3, 4	syl 14	. . 3 |- ( ( G ` (/) ) e. V -> <. (/) , ( G ` (/) ) >. e. { <. (/) , ( G ` (/) ) >. } )
6		fnsng 5265	. . . . 5 |- ( ( (/) e. _V /\ ( G ` (/) ) e. V ) -> { <. (/) , ( G ` (/) ) >. } Fn { (/) } )
7	1, 6	mpan 424	. . . 4 |- ( ( G ` (/) ) e. V -> { <. (/) , ( G ` (/) ) >. } Fn { (/) } )
8		fvsng 5715	. . . . . . 7 |- ( ( (/) e. _V /\ ( G ` (/) ) e. V ) -> ( { <. (/) , ( G ` (/) ) >. } ` (/) ) = ( G ` (/) ) )
9	1, 8	mpan 424	. . . . . 6 |- ( ( G ` (/) ) e. V -> ( { <. (/) , ( G ` (/) ) >. } ` (/) ) = ( G ` (/) ) )
10		res0 4913	. . . . . . 7 |- ( { <. (/) , ( G ` (/) ) >. } |` (/) ) = (/)
11	10	fveq2i 5520	. . . . . 6 |- ( G ` ( { <. (/) , ( G ` (/) ) >. } |` (/) ) ) = ( G ` (/) )
12	9, 11	eqtr4di 2228	. . . . 5 |- ( ( G ` (/) ) e. V -> ( { <. (/) , ( G ` (/) ) >. } ` (/) ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` (/) ) ) )
13		fveq2 5517	. . . . . . 7 |- ( y = (/) -> ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( { <. (/) , ( G ` (/) ) >. } ` (/) ) )
14		reseq2 4904	. . . . . . . 8 |- ( y = (/) -> ( { <. (/) , ( G ` (/) ) >. } |` y ) = ( { <. (/) , ( G ` (/) ) >. } |` (/) ) )
15	14	fveq2d 5521	. . . . . . 7 |- ( y = (/) -> ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` (/) ) ) )
16	13, 15	eqeq12d 2192	. . . . . 6 |- ( y = (/) -> ( ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) <-> ( { <. (/) , ( G ` (/) ) >. } ` (/) ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` (/) ) ) ) )
17	1, 16	ralsn 3637	. . . . 5 |- ( A. y e. { (/) } ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) <-> ( { <. (/) , ( G ` (/) ) >. } ` (/) ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` (/) ) ) )
18	12, 17	sylibr 134	. . . 4 |- ( ( G ` (/) ) e. V -> A. y e. { (/) } ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) )
19		suc0 4413	. . . . . 6 |- suc (/) = { (/) }
20		0elon 4394	. . . . . . 7 |- (/) e. On
21	20	onsuci 4517	. . . . . 6 |- suc (/) e. On
22	19, 21	eqeltrri 2251	. . . . 5 |- { (/) } e. On
23		fneq2 5307	. . . . . . 7 |- ( x = { (/) } -> ( { <. (/) , ( G ` (/) ) >. } Fn x <-> { <. (/) , ( G ` (/) ) >. } Fn { (/) } ) )
24		raleq 2673	. . . . . . 7 |- ( x = { (/) } -> ( A. y e. x ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) <-> A. y e. { (/) } ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) ) )
25	23, 24	anbi12d 473	. . . . . 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 2843	. . . . 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 424	. . . 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 411	. . 3 |- ( ( G ` (/) ) e. V -> E. x e. On ( { <. (/) , ( G ` (/) ) >. } Fn x /\ A. y e. x ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) ) )
29		snexg 4186	. . . . 5 |- ( <. (/) , ( G ` (/) ) >. e. _V -> { <. (/) , ( G ` (/) ) >. } e. _V )
30		eleq2 2241	. . . . . . 7 |- ( f = { <. (/) , ( G ` (/) ) >. } -> ( <. (/) , ( G ` (/) ) >. e. f <-> <. (/) , ( G ` (/) ) >. e. { <. (/) , ( G ` (/) ) >. } ) )
31		fneq1 5306	. . . . . . . . 9 |- ( f = { <. (/) , ( G ` (/) ) >. } -> ( f Fn x <-> { <. (/) , ( G ` (/) ) >. } Fn x ) )
32		fveq1 5516	. . . . . . . . . . 11 |- ( f = { <. (/) , ( G ` (/) ) >. } -> ( f ` y ) = ( { <. (/) , ( G ` (/) ) >. } ` y ) )
33		reseq1 4903	. . . . . . . . . . . 12 |- ( f = { <. (/) , ( G ` (/) ) >. } -> ( f |` y ) = ( { <. (/) , ( G ` (/) ) >. } |` y ) )
34	33	fveq2d 5521	. . . . . . . . . . 11 |- ( f = { <. (/) , ( G ` (/) ) >. } -> ( G ` ( f |` y ) ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) )
35	32, 34	eqeq12d 2192	. . . . . . . . . 10 |- ( f = { <. (/) , ( G ` (/) ) >. } -> ( ( f ` y ) = ( G ` ( f |` y ) ) <-> ( { <. (/) , ( G ` (/) ) >. } ` y ) = ( G ` ( { <. (/) , ( G ` (/) ) >. } |` y ) ) ) )
36	35	ralbidv 2477	. . . . . . . . 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 473	. . . . . . . 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 2478	. . . . . . 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 473	. . . . . 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 2827	. . . . 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 2244	. . . . 5 |- ( <. (/) , ( G ` (/) ) >. e. F <-> <. (/) , ( G ` (/) ) >. e. recs ( G ) )
44		df-recs 6309	. . . . . 6 |- recs ( G ) = U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
45	44	eleq2i 2244	. . . . 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 3823	. . . . 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 206	. . . 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	imbitrrdi 162	. . 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 433	. 2 |- ( ( G ` (/) ) e. V -> <. (/) , ( G ` (/) ) >. e. F )
50		opeldmg 4834	. . 3 |- ( ( (/) e. _V /\ ( G ` (/) ) e. V ) -> ( <. (/) , ( G ` (/) ) >. e. F -> (/) e. dom F ) )
51	1, 50	mpan 424	. 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 104   = wceq 1353   E.wex 1492   e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   _Vcvv 2739   (/)c0 3424   {csn 3594   <.cop 3597   U.cuni 3811   Oncon0 4365   suc csuc 4367   dom cdm 4628   |` cres 4630   Fn wfn 5213   ` cfv 5218  recscrecs 6308

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-recs 6309

This theorem is referenced by:  tfr0  6327