Theorem tfr0 5878

Description: Transfinite recursion at the empty set. (Contributed by Jim Kingdon, 8-May-2020.)

Hypothesis

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

Assertion

Ref	Expression
tfr0	|- (( G ` (/)) e. V -> ( F ` (/)) = ( G ` (/)))

Proof of Theorem tfr0

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

Step	Hyp	Ref	Expression
1		0ex 3875	. . . . 5 |- (/) e. _V
2		opexg 3955	. . . . 5 |- (( (/) e. _V /\ ( G ` (/)) e. V) -> <. (/) , ( G ` (/)) >. e. _V)
3	1, 2	mpan 400	. . . 4 |- (( G ` (/)) e. V -> <. (/) , ( G ` (/)) >. e. _V)
4		snidg 3392	. . . 4 |- ( <. (/) , ( G ` (/)) >. e. _V -> <. (/) , ( G ` (/)) >. e. { <. (/) , ( G ` (/)) >. })
5	3, 4	syl 14	. . 3 |- (( G ` (/)) e. V -> <. (/) , ( G ` (/)) >. e. { <. (/) , ( G ` (/)) >. })
6		fnsng 4890	. . . . 5 |- (( (/) e. _V /\ ( G ` (/)) e. V) -> { <. (/) , ( G ` (/)) >. } Fn { (/) })
7	1, 6	mpan 400	. . . 4 |- (( G ` (/)) e. V -> { <. (/) , ( G ` (/)) >. } Fn { (/) })
8		fvsng 5302	. . . . . . 7 |- (( (/) e. _V /\ ( G ` (/)) e. V) -> ( { <. (/) , ( G ` (/)) >. } ` (/)) = ( G ` (/)))
9	1, 8	mpan 400	. . . . . 6 |- (( G ` (/)) e. V -> ( { <. (/) , ( G ` (/)) >. } ` (/)) = ( G ` (/)))
10		res0 4559	. . . . . . 7 |- ( { <. (/) , ( G ` (/)) >. } |` (/)) = (/)
11	10	fveq2i 5124	. . . . . 6 |- ( G ` ( { <. (/) , ( G ` (/)) >. } |` (/))) = ( G ` (/))
12	9, 11	syl6eqr 2087	. . . . 5 |- (( G ` (/)) e. V -> ( { <. (/) , ( G ` (/)) >. } ` (/)) = ( G ` ( { <. (/) , ( G ` (/)) >. } |` (/))))
13		fveq2 5121	. . . . . . 7 |- (y = (/) -> ( { <. (/) , ( G ` (/)) >. } ` y) = ( { <. (/) , ( G ` (/)) >. } ` (/)))
14		reseq2 4550	. . . . . . . 8 |- (y = (/) -> ( { <. (/) , ( G ` (/)) >. } |` y) = ( { <. (/) , ( G ` (/)) >. } |` (/)))
15	14	fveq2d 5125	. . . . . . 7 |- (y = (/) -> ( G ` ( { <. (/) , ( G ` (/)) >. } |` y)) = ( G ` ( { <. (/) , ( G ` (/)) >. } |` (/))))
16	13, 15	eqeq12d 2051	. . . . . 6 |- (y = (/) -> (( { <. (/) , ( G ` (/)) >. } ` y) = ( G ` ( { <. (/) , ( G ` (/)) >. } |` y)) <-> ( { <. (/) , ( G ` (/)) >. } ` (/)) = ( G ` ( { <. (/) , ( G ` (/)) >. } |` (/)))))
17	1, 16	ralsn 3405	. . . . 5 |- (A.y e. { (/) } ( { <. (/) , ( G ` (/)) >. } ` y) = ( G ` ( { <. (/) , ( G ` (/)) >. } |` y)) <-> ( { <. (/) , ( G ` (/)) >. } ` (/)) = ( G ` ( { <. (/) , ( G ` (/)) >. } |` (/))))
18	12, 17	sylibr 137	. . . 4 |- (( G ` (/)) e. V -> A.y e. { (/) } ( { <. (/) , ( G ` (/)) >. } ` y) = ( G ` ( { <. (/) , ( G ` (/)) >. } |` y)))
19		suc0 4114	. . . . . 6 |- suc (/) = { (/) }
20		0elon 4095	. . . . . . 7 |- (/) e. On
21	20	onsuci 4207	. . . . . 6 |- suc (/) e. On
22	19, 21	eqeltrri 2108	. . . . 5 |- { (/) } e. On
23		fneq2 4931	. . . . . . 7 |- (x = { (/) } -> ( { <. (/) , ( G ` (/)) >. } Fn x <-> { <. (/) , ( G ` (/)) >. } Fn { (/) }))
24		raleq 2499	. . . . . . 7 |- (x = { (/) } -> (A.y e. x ( { <. (/) , ( G ` (/)) >. } ` y) = ( G ` ( { <. (/) , ( G ` (/)) >. } |` y)) <-> A.y e. { (/) } ( { <. (/) , ( G ` (/)) >. } ` y) = ( G ` ( { <. (/) , ( G ` (/)) >. } |` y))))
25	23, 24	anbi12d 442	. . . . . 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 2650	. . . . 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 400	. . . 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 391	. . 3 |- (( G ` (/)) e. V -> E.x e. On ( { <. (/) , ( G ` (/)) >. } Fn x /\ A.y e. x ( { <. (/) , ( G ` (/)) >. } ` y) = ( G ` ( { <. (/) , ( G ` (/)) >. } |` y))))
29		snexg 3927	. . . . 5 |- ( <. (/) , ( G ` (/)) >. e. _V -> { <. (/) , ( G ` (/)) >. } e. _V)
30		eleq2 2098	. . . . . . 7 |- (f = { <. (/) , ( G ` (/)) >. } -> ( <. (/) , ( G ` (/)) >. e. f <-> <. (/) , ( G ` (/)) >. e. { <. (/) , ( G ` (/)) >. }))
31		fneq1 4930	. . . . . . . . 9 |- (f = { <. (/) , ( G ` (/)) >. } -> (f Fn x <-> { <. (/) , ( G ` (/)) >. } Fn x))
32		fveq1 5120	. . . . . . . . . . 11 |- (f = { <. (/) , ( G ` (/)) >. } -> (f ` y) = ( { <. (/) , ( G ` (/)) >. } ` y))
33		reseq1 4549	. . . . . . . . . . . 12 |- (f = { <. (/) , ( G ` (/)) >. } -> (f |` y) = ( { <. (/) , ( G ` (/)) >. } |` y))
34	33	fveq2d 5125	. . . . . . . . . . 11 |- (f = { <. (/) , ( G ` (/)) >. } -> ( G ` (f |` y)) = ( G ` ( { <. (/) , ( G ` (/)) >. } |` y)))
35	32, 34	eqeq12d 2051	. . . . . . . . . 10 |- (f = { <. (/) , ( G ` (/)) >. } -> ((f ` y) = ( G ` (f |` y)) <-> ( { <. (/) , ( G ` (/)) >. } ` y) = ( G ` ( { <. (/) , ( G ` (/)) >. } |` y))))
36	35	ralbidv 2320	. . . . . . . . 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 442	. . . . . . . 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 2321	. . . . . . 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 442	. . . . . 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 2635	. . . . 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 2101	. . . . 5 |- ( <. (/) , ( G ` (/)) >. e. F <-> <. (/) , ( G ` (/)) >. e. recs( G))
44		df-recs 5861	. . . . . 6 |- recs( G) = U. {f | E.x e. On (f Fn x /\ A.y e. x (f ` y) = ( G ` (f |` y))) }
45	44	eleq2i 2101	. . . . 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 3583	. . . . 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 195	. . . 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 151	. . 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 409	. 2 |- (( G ` (/)) e. V -> <. (/) , ( G ` (/)) >. e. F)
50		opeldmg 4483	. . . . 5 |- (( (/) e. _V /\ ( G ` (/)) e. V) -> ( <. (/) , ( G ` (/)) >. e. F -> (/) e. dom F))
51	1, 50	mpan 400	. . . 4 |- (( G ` (/)) e. V -> ( <. (/) , ( G ` (/)) >. e. F -> (/) e. dom F))
52	42	tfr2a 5877	. . . 4 |- ( (/) e. dom F -> ( F ` (/)) = ( G ` ( F |` (/))))
53	51, 52	syl6 29	. . 3 |- (( G ` (/)) e. V -> ( <. (/) , ( G ` (/)) >. e. F -> ( F ` (/)) = ( G ` ( F |` (/)))))
54		res0 4559	. . . . 5 |- ( F |` (/)) = (/)
55	54	fveq2i 5124	. . . 4 |- ( G ` ( F |` (/))) = ( G ` (/))
56	55	eqeq2i 2047	. . 3 |- (( F ` (/)) = ( G ` ( F |` (/))) <-> ( F ` (/)) = ( G ` (/)))
57	53, 56	syl6ib 150	. 2 |- (( G ` (/)) e. V -> ( <. (/) , ( G ` (/)) >. e. F -> ( F ` (/)) = ( G ` (/))))
58	49, 57	mpd 13	1 |- (( G ` (/)) e. V -> ( F ` (/)) = ( G ` (/)))

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1242  E.wex 1378   e. wcel 1390   {cab 2023  A.wral 2300  E.wrex 2301   _Vcvv 2551   (/)c0 3218   {csn 3367   <.cop 3370   U.cuni 3571   Oncon0 4066   suc csuc 4068   dom cdm 4288   |` cres 4290   Fn wfn 4840   ` cfv 4845  recscrecs 5860

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-recs 5861

This theorem is referenced by:  rdg0  5914  frec0g  5922