Theorem tfri1dALT 6582

Description: Alternate proof of tfri1d 6566 in terms of tfr1on 6581.

Although this does show that the tfr1on 6581 proof is general enough to also prove tfri1d 6566, the tfri1d 6566 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 6551	. . . 4 |- Fun recs ( G )
2		tfri1dALT.1	. . . . 5 |- F = recs ( G )
3	2	funeqi 5373	. . . 4 |- ( Fun F <-> Fun recs ( G ) )
4	1, 3	mpbir 146	. . 3 |- Fun F
5	4	a1i 9	. 2 |- ( ph -> Fun F )
6		eqid 2232	. . . . . 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 6549	. . . . 5 |- Ord dom recs ( G )
8	2	dmeqi 4957	. . . . . 6 |- dom F = dom recs ( G )
9		ordeq 4493	. . . . . 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 4614	. . . 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 1504	. . . . . . . . . 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 1607	. . . . . . . 8 |- ( ph -> Fun G )
19	18	adantr 276	. . . . . . 7 |- ( ( ph /\ z e. On ) -> Fun G )
20		ordon 4608	. . . . . . . 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 1504	. . . . . . . . . 10 |- ( A. x ( Fun G /\ ( G ` x ) e. _V ) -> A. x ( G ` x ) e. _V )
24		fveq2 5670	. . . . . . . . . . . 12 |- ( x = f -> ( G ` x ) = ( G ` f ) )
25	24	eleq1d 2301	. . . . . . . . . . 11 |- ( x = f -> ( ( G ` x ) e. _V <-> ( G ` f ) e. _V ) )
26	25	spv 1909	. . . . . . . . . 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 1045	. . . . . . 7 |- ( ( ( ph /\ z e. On ) /\ y e. On /\ f Fn y ) -> ( G ` f ) e. _V )
30		onsuc 4623	. . . . . . . . 9 |- ( y e. On -> suc y e. On )
31		unon 4633	. . . . . . . . 9 |- U. On = On
32	30, 31	eleq2s 2327	. . . . . . . 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 4623	. . . . . . . 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 6581	. . . . . 6 |- ( ( ph /\ z e. On ) -> suc z C_ dom F )
37		vex 2816	. . . . . . 7 |- z e. _V
38	37	sucid 4538	. . . . . 6 |- z e. suc z
39		ssel2 3233	. . . . . 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 3244	. . 3 |- ( ph -> On C_ dom F )
43	13, 42	eqssd 3255	. 2 |- ( ph -> dom F = On )
44		df-fn 5355	. 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 1396   = wceq 1398   e. wcel 2203   {cab 2218   A.wral 2520   E.wrex 2521   _Vcvv 2813   C_ wss 3211   U.cuni 3914   Ord word 4483   Oncon0 4484   suc csuc 4486   dom cdm 4749   |` cres 4751   Fun wfun 5346   Fn wfn 5347   ` cfv 5352  recscrecs 6535

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-recs 6536

This theorem is referenced by: (None)