Theorem tfri1dALT 6295

Description: Alternate proof of tfri1d 6279 in terms of tfr1on 6294.

Although this does show that the tfr1on 6294 proof is general enough to also prove tfri1d 6279, the tfri1d 6279 proof is simpler in places because it does not need to deal with X being any ordinal. For that reason, we have both proofs. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Jim Kingdon, 20-Mar-2022.)

Hypotheses

Ref	Expression
tfri1dALT.1	|- F = recs ( G )
tfri1dALT.2	|- ( ph -> A. x ( Fun G /\ ( G ` x ) e. _V ) )

Assertion

Ref	Expression
tfri1dALT	|- ( ph -> F Fn On )

Distinct variable group:   x, G

Allowed substitution hints:   ph( x)   F( x)

Proof of Theorem tfri1dALT

Dummy variables z a b c f y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfrfun 6264	. . . 4 |- Fun recs ( G )
2		tfri1dALT.1	. . . . 5 |- F = recs ( G )
3	2	funeqi 5190	. . . 4 |- ( Fun F <-> Fun recs ( G ) )
4	1, 3	mpbir 145	. . 3 |- Fun F
5	4	a1i 9	. 2 |- ( ph -> Fun F )
6		eqid 2157	. . . . . 6 |- { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( G ` ( a |` c ) ) ) } = { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( G ` ( a |` c ) ) ) }
7	6	tfrlem8 6262	. . . . 5 |- Ord dom recs ( G )
8	2	dmeqi 4786	. . . . . 6 |- dom F = dom recs ( G )
9		ordeq 4332	. . . . . 6 |- ( dom F = dom recs ( G ) -> ( Ord dom F <-> Ord dom recs ( G ) ) )
10	8, 9	ax-mp 5	. . . . 5 |- ( Ord dom F <-> Ord dom recs ( G ) )
11	7, 10	mpbir 145	. . . 4 |- Ord dom F
12		ordsson 4450	. . . 4 |- ( Ord dom F -> dom F C_ On )
13	11, 12	mp1i 10	. . 3 |- ( ph -> dom F C_ On )
14		tfri1dALT.2	. . . . . . . . . 10 |- ( ph -> A. x ( Fun G /\ ( G ` x ) e. _V ) )
15		simpl 108	. . . . . . . . . . 11 |- ( ( Fun G /\ ( G ` x ) e. _V ) -> Fun G )
16	15	alimi 1435	. . . . . . . . . 10 |- ( A. x ( Fun G /\ ( G ` x ) e. _V ) -> A. x Fun G )
17	14, 16	syl 14	. . . . . . . . 9 |- ( ph -> A. x Fun G )
18	17	19.21bi 1538	. . . . . . . 8 |- ( ph -> Fun G )
19	18	adantr 274	. . . . . . 7 |- ( ( ph /\ z e. On ) -> Fun G )
20		ordon 4444	. . . . . . . 8 |- Ord On
21	20	a1i 9	. . . . . . 7 |- ( ( ph /\ z e. On ) -> Ord On )
22		simpr 109	. . . . . . . . . . 11 |- ( ( Fun G /\ ( G ` x ) e. _V ) -> ( G ` x ) e. _V )
23	22	alimi 1435	. . . . . . . . . 10 |- ( A. x ( Fun G /\ ( G ` x ) e. _V ) -> A. x ( G ` x ) e. _V )
24		fveq2 5467	. . . . . . . . . . . 12 |- ( x = f -> ( G ` x ) = ( G ` f ) )
25	24	eleq1d 2226	. . . . . . . . . . 11 |- ( x = f -> ( ( G ` x ) e. _V <-> ( G ` f ) e. _V ) )
26	25	spv 1840	. . . . . . . . . 10 |- ( A. x ( G ` x ) e. _V -> ( G ` f ) e. _V )
27	14, 23, 26	3syl 17	. . . . . . . . 9 |- ( ph -> ( G ` f ) e. _V )
28	27	adantr 274	. . . . . . . 8 |- ( ( ph /\ z e. On ) -> ( G ` f ) e. _V )
29	28	3ad2ant1 1003	. . . . . . 7 |- ( ( ( ph /\ z e. On ) /\ y e. On /\ f Fn y ) -> ( G ` f ) e. _V )
30		suceloni 4459	. . . . . . . . 9 |- ( y e. On -> suc y e. On )
31		unon 4469	. . . . . . . . 9 |- U. On = On
32	30, 31	eleq2s 2252	. . . . . . . 8 |- ( y e. U. On -> suc y e. On )
33	32	adantl 275	. . . . . . 7 |- ( ( ( ph /\ z e. On ) /\ y e. U. On ) -> suc y e. On )
34		suceloni 4459	. . . . . . . 8 |- ( z e. On -> suc z e. On )
35	34	adantl 275	. . . . . . 7 |- ( ( ph /\ z e. On ) -> suc z e. On )
36	2, 19, 21, 29, 33, 35	tfr1on 6294	. . . . . 6 |- ( ( ph /\ z e. On ) -> suc z C_ dom F )
37		vex 2715	. . . . . . 7 |- z e. _V
38	37	sucid 4377	. . . . . 6 |- z e. suc z
39		ssel2 3123	. . . . . 6 |- ( ( suc z C_ dom F /\ z e. suc z ) -> z e. dom F )
40	36, 38, 39	sylancl 410	. . . . 5 |- ( ( ph /\ z e. On ) -> z e. dom F )
41	40	ex 114	. . . 4 |- ( ph -> ( z e. On -> z e. dom F ) )
42	41	ssrdv 3134	. . 3 |- ( ph -> On C_ dom F )
43	13, 42	eqssd 3145	. 2 |- ( ph -> dom F = On )
44		df-fn 5172	. 2 |- ( F Fn On <-> ( Fun F /\ dom F = On ) )
45	5, 43, 44	sylanbrc 414	1 |- ( ph -> F Fn On )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1333   = wceq 1335   e. wcel 2128   {cab 2143   A.wral 2435   E.wrex 2436   _Vcvv 2712   C_ wss 3102   U.cuni 3772   Ord word 4322   Oncon0 4323   suc csuc 4325   dom cdm 4585   |` cres 4587   Fun wfun 5163   Fn wfn 5164   ` cfv 5169  recscrecs 6248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-recs 6249

This theorem is referenced by: (None)