Theorem tfrlem16 7489

Description: Lemma for finite recursion. Without assuming ax-rep 4771, we can show that the domain of the constructed function is a limit ordinal, and hence contains all the finite ordinals. (Contributed by Mario Carneiro, 14-Nov-2014.)

Hypothesis

Ref	Expression
tfrlem.1	|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }

Assertion

Ref	Expression
tfrlem16	|- Lim dom recs ( F )

Distinct variable group:   x, f, y, F

Allowed substitution hints:   A( x, y, f)

Proof of Theorem tfrlem16

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfrlem.1	. . . 4 |- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
2	1	tfrlem8 7480	. . 3 |- Ord dom recs ( F )
3		ordzsl 7045	. . 3 |- ( Ord dom recs ( F ) <-> ( dom recs ( F ) = (/) \/ E. z e. On dom recs ( F ) = suc z \/ Lim dom recs ( F ) ) )
4	2, 3	mpbi 220	. 2 |- ( dom recs ( F ) = (/) \/ E. z e. On dom recs ( F ) = suc z \/ Lim dom recs ( F ) )
5		res0 5400	. . . . . . 7 |- (recs ( F ) |` (/) ) = (/)
6		0ex 4790	. . . . . . 7 |- (/) e. _V
7	5, 6	eqeltri 2697	. . . . . 6 |- (recs ( F ) |` (/) ) e. _V
8		0elon 5778	. . . . . . 7 |- (/) e. On
9	1	tfrlem15 7488	. . . . . . 7 |- ( (/) e. On -> ( (/) e. dom recs ( F ) <-> (recs ( F ) |` (/) ) e. _V ) )
10	8, 9	ax-mp 5	. . . . . 6 |- ( (/) e. dom recs ( F ) <-> (recs ( F ) |` (/) ) e. _V )
11	7, 10	mpbir 221	. . . . 5 |- (/) e. dom recs ( F )
12	11	n0ii 3922	. . . 4 |- -. dom recs ( F ) = (/)
13	12	pm2.21i 116	. . 3 |- ( dom recs ( F ) = (/) -> Lim dom recs ( F ) )
14	1	tfrlem13 7486	. . . . 5 |- -. recs ( F ) e. _V
15		simpr 477	. . . . . . . . . 10 |- ( ( z e. On /\ dom recs ( F ) = suc z ) -> dom recs ( F ) = suc z )
16		df-suc 5729	. . . . . . . . . 10 |- suc z = ( z u. { z } )
17	15, 16	syl6eq 2672	. . . . . . . . 9 |- ( ( z e. On /\ dom recs ( F ) = suc z ) -> dom recs ( F ) = ( z u. { z } ) )
18	17	reseq2d 5396	. . . . . . . 8 |- ( ( z e. On /\ dom recs ( F ) = suc z ) -> (recs ( F ) |` dom recs ( F ) ) = (recs ( F ) |` ( z u. { z } ) ) )
19	1	tfrlem6 7478	. . . . . . . . 9 |- Rel recs ( F )
20		resdm 5441	. . . . . . . . 9 |- ( Rel recs ( F ) -> (recs ( F ) |` dom recs ( F ) ) = recs ( F ) )
21	19, 20	ax-mp 5	. . . . . . . 8 |- (recs ( F ) |` dom recs ( F ) ) = recs ( F )
22		resundi 5410	. . . . . . . 8 |- (recs ( F ) |` ( z u. { z } ) ) = ( (recs ( F ) |` z ) u. (recs ( F ) |` { z } ) )
23	18, 21, 22	3eqtr3g 2679	. . . . . . 7 |- ( ( z e. On /\ dom recs ( F ) = suc z ) -> recs ( F ) = ( (recs ( F ) |` z ) u. (recs ( F ) |` { z } ) ) )
24		vex 3203	. . . . . . . . . . 11 |- z e. _V
25	24	sucid 5804	. . . . . . . . . 10 |- z e. suc z
26	25, 15	syl5eleqr 2708	. . . . . . . . 9 |- ( ( z e. On /\ dom recs ( F ) = suc z ) -> z e. dom recs ( F ) )
27	1	tfrlem9a 7482	. . . . . . . . 9 |- ( z e. dom recs ( F ) -> (recs ( F ) |` z ) e. _V )
28	26, 27	syl 17	. . . . . . . 8 |- ( ( z e. On /\ dom recs ( F ) = suc z ) -> (recs ( F ) |` z ) e. _V )
29		snex 4908	. . . . . . . . 9 |- { <. z , (recs ( F ) ` z ) >. } e. _V
30	1	tfrlem7 7479	. . . . . . . . . 10 |- Fun recs ( F )
31		funressn 6426	. . . . . . . . . 10 |- ( Fun recs ( F ) -> (recs ( F ) |` { z } ) C_ { <. z , (recs ( F ) ` z ) >. } )
32	30, 31	ax-mp 5	. . . . . . . . 9 |- (recs ( F ) |` { z } ) C_ { <. z , (recs ( F ) ` z ) >. }
33	29, 32	ssexi 4803	. . . . . . . 8 |- (recs ( F ) |` { z } ) e. _V
34		unexg 6959	. . . . . . . 8 |- ( ( (recs ( F ) |` z ) e. _V /\ (recs ( F ) |` { z } ) e. _V ) -> ( (recs ( F ) |` z ) u. (recs ( F ) |` { z } ) ) e. _V )
35	28, 33, 34	sylancl 694	. . . . . . 7 |- ( ( z e. On /\ dom recs ( F ) = suc z ) -> ( (recs ( F ) |` z ) u. (recs ( F ) |` { z } ) ) e. _V )
36	23, 35	eqeltrd 2701	. . . . . 6 |- ( ( z e. On /\ dom recs ( F ) = suc z ) -> recs ( F ) e. _V )
37	36	rexlimiva 3028	. . . . 5 |- ( E. z e. On dom recs ( F ) = suc z -> recs ( F ) e. _V )
38	14, 37	mto 188	. . . 4 |- -. E. z e. On dom recs ( F ) = suc z
39	38	pm2.21i 116	. . 3 |- ( E. z e. On dom recs ( F ) = suc z -> Lim dom recs ( F ) )
40		id 22	. . 3 |- ( Lim dom recs ( F ) -> Lim dom recs ( F ) )
41	13, 39, 40	3jaoi 1391	. 2 |- ( ( dom recs ( F ) = (/) \/ E. z e. On dom recs ( F ) = suc z \/ Lim dom recs ( F ) ) -> Lim dom recs ( F ) )
42	4, 41	ax-mp 5	1 |- Lim dom recs ( F )

Syntax hints:   <-> wb 196   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   dom cdm 5114   |` cres 5116   Rel wrel 5119   Ord word 5722   Oncon0 5723   Lim wlim 5724   suc csuc 5725   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  recscrecs 7467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-wrecs 7407  df-recs 7468

This theorem is referenced by:  tfr1a  7490