Theorem tfri1dALT 6370

Description: Alternate proof of tfri1d 6354 in terms of tfr1on 6369.

Although this does show that the tfr1on 6369 proof is general enough to also prove tfri1d 6354, the tfri1d 6354 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 6339	. . . 4 |- Fun recs ( G )
2		tfri1dALT.1	. . . . 5 |- F = recs ( G )
3	2	funeqi 5252	. . . 4 |- ( Fun F <-> Fun recs ( G ) )
4	1, 3	mpbir 146	. . 3 |- Fun F
5	4	a1i 9	. 2 |- ( ph -> Fun F )
6		eqid 2189	. . . . . 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 6337	. . . . 5 |- Ord dom recs ( G )
8	2	dmeqi 4843	. . . . . 6 |- dom F = dom recs ( G )
9		ordeq 4387	. . . . . 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 4506	. . . 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 1466	. . . . . . . . . 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 1569	. . . . . . . 8 |- ( ph -> Fun G )
19	18	adantr 276	. . . . . . 7 |- ( ( ph /\ z e. On ) -> Fun G )
20		ordon 4500	. . . . . . . 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 1466	. . . . . . . . . 10 |- ( A. x ( Fun G /\ ( G ` x ) e. _V ) -> A. x ( G ` x ) e. _V )
24		fveq2 5530	. . . . . . . . . . . 12 |- ( x = f -> ( G ` x ) = ( G ` f ) )
25	24	eleq1d 2258	. . . . . . . . . . 11 |- ( x = f -> ( ( G ` x ) e. _V <-> ( G ` f ) e. _V ) )
26	25	spv 1871	. . . . . . . . . 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 4515	. . . . . . . . 9 |- ( y e. On -> suc y e. On )
31		unon 4525	. . . . . . . . 9 |- U. On = On
32	30, 31	eleq2s 2284	. . . . . . . 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 4515	. . . . . . . 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 6369	. . . . . 6 |- ( ( ph /\ z e. On ) -> suc z C_ dom F )
37		vex 2755	. . . . . . 7 |- z e. _V
38	37	sucid 4432	. . . . . 6 |- z e. suc z
39		ssel2 3165	. . . . . 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 3176	. . 3 |- ( ph -> On C_ dom F )
43	13, 42	eqssd 3187	. 2 |- ( ph -> dom F = On )
44		df-fn 5234	. 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 1362   = wceq 1364   e. wcel 2160   {cab 2175   A.wral 2468   E.wrex 2469   _Vcvv 2752   C_ wss 3144   U.cuni 3824   Ord word 4377   Oncon0 4378   suc csuc 4380   dom cdm 4641   |` cres 4643   Fun wfun 5225   Fn wfn 5226   ` cfv 5231  recscrecs 6323

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-recs 6324

This theorem is referenced by: (None)