Theorem tfri1dALT 6436

Description: Alternate proof of tfri1d 6420 in terms of tfr1on 6435.

Although this does show that the tfr1on 6435 proof is general enough to also prove tfri1d 6420, the tfri1d 6420 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 6405	. . . 4 |- Fun recs ( G )
2		tfri1dALT.1	. . . . 5 |- F = recs ( G )
3	2	funeqi 5291	. . . 4 |- ( Fun F <-> Fun recs ( G ) )
4	1, 3	mpbir 146	. . 3 |- Fun F
5	4	a1i 9	. 2 |- ( ph -> Fun F )
6		eqid 2204	. . . . . 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 6403	. . . . 5 |- Ord dom recs ( G )
8	2	dmeqi 4878	. . . . . 6 |- dom F = dom recs ( G )
9		ordeq 4418	. . . . . 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 4539	. . . 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 1477	. . . . . . . . . 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 1580	. . . . . . . 8 |- ( ph -> Fun G )
19	18	adantr 276	. . . . . . 7 |- ( ( ph /\ z e. On ) -> Fun G )
20		ordon 4533	. . . . . . . 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 1477	. . . . . . . . . 10 |- ( A. x ( Fun G /\ ( G ` x ) e. _V ) -> A. x ( G ` x ) e. _V )
24		fveq2 5575	. . . . . . . . . . . 12 |- ( x = f -> ( G ` x ) = ( G ` f ) )
25	24	eleq1d 2273	. . . . . . . . . . 11 |- ( x = f -> ( ( G ` x ) e. _V <-> ( G ` f ) e. _V ) )
26	25	spv 1882	. . . . . . . . . 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 1020	. . . . . . 7 |- ( ( ( ph /\ z e. On ) /\ y e. On /\ f Fn y ) -> ( G ` f ) e. _V )
30		onsuc 4548	. . . . . . . . 9 |- ( y e. On -> suc y e. On )
31		unon 4558	. . . . . . . . 9 |- U. On = On
32	30, 31	eleq2s 2299	. . . . . . . 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 4548	. . . . . . . 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 6435	. . . . . 6 |- ( ( ph /\ z e. On ) -> suc z C_ dom F )
37		vex 2774	. . . . . . 7 |- z e. _V
38	37	sucid 4463	. . . . . 6 |- z e. suc z
39		ssel2 3187	. . . . . 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 3198	. . 3 |- ( ph -> On C_ dom F )
43	13, 42	eqssd 3209	. 2 |- ( ph -> dom F = On )
44		df-fn 5273	. 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 1370   = wceq 1372   e. wcel 2175   {cab 2190   A.wral 2483   E.wrex 2484   _Vcvv 2771   C_ wss 3165   U.cuni 3849   Ord word 4408   Oncon0 4409   suc csuc 4411   dom cdm 4674   |` cres 4676   Fun wfun 5264   Fn wfn 5265   ` cfv 5270  recscrecs 6389

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-iord 4412  df-on 4414  df-suc 4417  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-recs 6390

This theorem is referenced by: (None)