Theorem tfrlem16 8029

Description: Lemma for finite recursion. Without assuming ax-rep 5190, 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 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 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 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
2	1	tfrlem8 8020	. . 3 ⊢ Ord dom recs(𝐹)
3		ordzsl 7560	. . 3 ⊢ (Ord dom recs(𝐹) ↔ (dom recs(𝐹) = ∅ ∨ ∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧 ∨ Lim dom recs(𝐹)))
4	2, 3	mpbi 232	. 2 ⊢ (dom recs(𝐹) = ∅ ∨ ∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧 ∨ Lim dom recs(𝐹))
5		res0 5857	. . . . . . 7 ⊢ (recs(𝐹) ↾ ∅) = ∅
6		0ex 5211	. . . . . . 7 ⊢ ∅ ∈ V
7	5, 6	eqeltri 2909	. . . . . 6 ⊢ (recs(𝐹) ↾ ∅) ∈ V
8		0elon 6244	. . . . . . 7 ⊢ ∅ ∈ On
9	1	tfrlem15 8028	. . . . . . 7 ⊢ (∅ ∈ On → (∅ ∈ dom recs(𝐹) ↔ (recs(𝐹) ↾ ∅) ∈ V))
10	8, 9	ax-mp 5	. . . . . 6 ⊢ (∅ ∈ dom recs(𝐹) ↔ (recs(𝐹) ↾ ∅) ∈ V)
11	7, 10	mpbir 233	. . . . 5 ⊢ ∅ ∈ dom recs(𝐹)
12	11	n0ii 4302	. . . 4 ⊢ ¬ dom recs(𝐹) = ∅
13	12	pm2.21i 119	. . 3 ⊢ (dom recs(𝐹) = ∅ → Lim dom recs(𝐹))
14	1	tfrlem13 8026	. . . . 5 ⊢ ¬ recs(𝐹) ∈ V
15		simpr 487	. . . . . . . . . 10 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → dom recs(𝐹) = suc 𝑧)
16		df-suc 6197	. . . . . . . . . 10 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧})
17	15, 16	syl6eq 2872	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → dom recs(𝐹) = (𝑧 ∪ {𝑧}))
18	17	reseq2d 5853	. . . . . . . 8 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → (recs(𝐹) ↾ dom recs(𝐹)) = (recs(𝐹) ↾ (𝑧 ∪ {𝑧})))
19	1	tfrlem6 8018	. . . . . . . . 9 ⊢ Rel recs(𝐹)
20		resdm 5897	. . . . . . . . 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 5867	. . . . . . . 8 ⊢ (recs(𝐹) ↾ (𝑧 ∪ {𝑧})) = ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧}))
23	18, 21, 22	3eqtr3g 2879	. . . . . . 7 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → recs(𝐹) = ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})))
24		vex 3497	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
25	24	sucid 6270	. . . . . . . . . 10 ⊢ 𝑧 ∈ suc 𝑧
26	25, 15	eleqtrrid 2920	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → 𝑧 ∈ dom recs(𝐹))
27	1	tfrlem9a 8022	. . . . . . . . 9 ⊢ (𝑧 ∈ dom recs(𝐹) → (recs(𝐹) ↾ 𝑧) ∈ V)
28	26, 27	syl 17	. . . . . . . 8 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → (recs(𝐹) ↾ 𝑧) ∈ V)
29		snex 5332	. . . . . . . . 9 ⊢ {⟨𝑧, (recs(𝐹)‘𝑧)⟩} ∈ V
30	1	tfrlem7 8019	. . . . . . . . . 10 ⊢ Fun recs(𝐹)
31		funressn 6921	. . . . . . . . . 10 ⊢ (Fun recs(𝐹) → (recs(𝐹) ↾ {𝑧}) ⊆ {⟨𝑧, (recs(𝐹)‘𝑧)⟩})
32	30, 31	ax-mp 5	. . . . . . . . 9 ⊢ (recs(𝐹) ↾ {𝑧}) ⊆ {⟨𝑧, (recs(𝐹)‘𝑧)⟩}
33	29, 32	ssexi 5226	. . . . . . . 8 ⊢ (recs(𝐹) ↾ {𝑧}) ∈ V
34		unexg 7472	. . . . . . . 8 ⊢ (((recs(𝐹) ↾ 𝑧) ∈ V ∧ (recs(𝐹) ↾ {𝑧}) ∈ V) → ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})) ∈ V)
35	28, 33, 34	sylancl 588	. . . . . . 7 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})) ∈ V)
36	23, 35	eqeltrd 2913	. . . . . 6 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → recs(𝐹) ∈ V)
37	36	rexlimiva 3281	. . . . 5 ⊢ (∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧 → recs(𝐹) ∈ V)
38	14, 37	mto 199	. . . 4 ⊢ ¬ ∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧
39	38	pm2.21i 119	. . 3 ⊢ (∃𝑧 ∈ 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 1423	. 2 ⊢ ((dom recs(𝐹) = ∅ ∨ ∃𝑧 ∈ 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 208   ∧ wa 398   ∨ w3o 1082   = wceq 1537   ∈ wcel 2114  {cab 2799  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ∪ cun 3934   ⊆ wss 3936  ∅c0 4291  {csn 4567  ⟨cop 4573  dom cdm 5555   ↾ cres 5557  Rel wrel 5560  Ord word 6190  Oncon0 6191  Lim wlim 6192  suc csuc 6193  Fun wfun 6349   Fn wfn 6350  ‘cfv 6355  recscrecs 8007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-wrecs 7947  df-recs 8008

This theorem is referenced by:  tfr1a  8030