Theorem tfri1dALT 6512

Description: Alternate proof of tfri1d 6496 in terms of tfr1on 6511.

Although this does show that the tfr1on 6511 proof is general enough to also prove tfri1d 6496, the tfri1d 6496 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 6481	. . . 4 |- Fun recs ( G )
2		tfri1dALT.1	. . . . 5 |- F = recs ( G )
3	2	funeqi 5345	. . . 4 |- ( Fun F <-> Fun recs ( G ) )
4	1, 3	mpbir 146	. . 3 |- Fun F
5	4	a1i 9	. 2 |- ( ph -> Fun F )
6		eqid 2229	. . . . . 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 6479	. . . . 5 |- Ord dom recs ( G )
8	2	dmeqi 4930	. . . . . 6 |- dom F = dom recs ( G )
9		ordeq 4467	. . . . . 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 146	. . . 4 |- Ord dom F
12		ordsson 4588	. . . 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 109	. . . . . . . . . . 11 |- ( ( Fun G /\ ( G ` x ) e. _V ) -> Fun G )
16	15	alimi 1501	. . . . . . . . . 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 1604	. . . . . . . 8 |- ( ph -> Fun G )
19	18	adantr 276	. . . . . . 7 |- ( ( ph /\ z e. On ) -> Fun G )
20		ordon 4582	. . . . . . . 8 |- Ord On
21	20	a1i 9	. . . . . . 7 |- ( ( ph /\ z e. On ) -> Ord On )
22		simpr 110	. . . . . . . . . . 11 |- ( ( Fun G /\ ( G ` x ) e. _V ) -> ( G ` x ) e. _V )
23	22	alimi 1501	. . . . . . . . . 10 |- ( A. x ( Fun G /\ ( G ` x ) e. _V ) -> A. x ( G ` x ) e. _V )
24		fveq2 5635	. . . . . . . . . . . 12 |- ( x = f -> ( G ` x ) = ( G ` f ) )
25	24	eleq1d 2298	. . . . . . . . . . 11 |- ( x = f -> ( ( G ` x ) e. _V <-> ( G ` f ) e. _V ) )
26	25	spv 1906	. . . . . . . . . 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 276	. . . . . . . 8 |- ( ( ph /\ z e. On ) -> ( G ` f ) e. _V )
29	28	3ad2ant1 1042	. . . . . . 7 |- ( ( ( ph /\ z e. On ) /\ y e. On /\ f Fn y ) -> ( G ` f ) e. _V )
30		onsuc 4597	. . . . . . . . 9 |- ( y e. On -> suc y e. On )
31		unon 4607	. . . . . . . . 9 |- U. On = On
32	30, 31	eleq2s 2324	. . . . . . . 8 |- ( y e. U. On -> suc y e. On )
33	32	adantl 277	. . . . . . 7 |- ( ( ( ph /\ z e. On ) /\ y e. U. On ) -> suc y e. On )
34		onsuc 4597	. . . . . . . 8 |- ( z e. On -> suc z e. On )
35	34	adantl 277	. . . . . . 7 |- ( ( ph /\ z e. On ) -> suc z e. On )
36	2, 19, 21, 29, 33, 35	tfr1on 6511	. . . . . 6 |- ( ( ph /\ z e. On ) -> suc z C_ dom F )
37		vex 2803	. . . . . . 7 |- z e. _V
38	37	sucid 4512	. . . . . 6 |- z e. suc z
39		ssel2 3220	. . . . . 6 |- ( ( suc z C_ dom F /\ z e. suc z ) -> z e. dom F )
40	36, 38, 39	sylancl 413	. . . . 5 |- ( ( ph /\ z e. On ) -> z e. dom F )
41	40	ex 115	. . . 4 |- ( ph -> ( z e. On -> z e. dom F ) )
42	41	ssrdv 3231	. . 3 |- ( ph -> On C_ dom F )
43	13, 42	eqssd 3242	. 2 |- ( ph -> dom F = On )
44		df-fn 5327	. 2 |- ( F Fn On <-> ( Fun F /\ dom F = On ) )
45	5, 43, 44	sylanbrc 417	1 |- ( ph -> F Fn On )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1393   = wceq 1395   e. wcel 2200   {cab 2215   A.wral 2508   E.wrex 2509   _Vcvv 2800   C_ wss 3198   U.cuni 3891   Ord word 4457   Oncon0 4458   suc csuc 4460   dom cdm 4723   |` cres 4725   Fun wfun 5318   Fn wfn 5319   ` cfv 5324  recscrecs 6465

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-recs 6466

This theorem is referenced by: (None)