Theorem tfrlem16 8338

Description: Lemma for finite recursion. Without assuming ax-rep 5229, 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 8329	. . 3 ⊢ Ord dom recs(𝐹)
3		ordzsl 7801	. . 3 ⊢ (Ord dom recs(𝐹) ↔ (dom recs(𝐹) = ∅ ∨ ∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧 ∨ Lim dom recs(𝐹)))
4	2, 3	mpbi 230	. 2 ⊢ (dom recs(𝐹) = ∅ ∨ ∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧 ∨ Lim dom recs(𝐹))
5		res0 5943	. . . . . . 7 ⊢ (recs(𝐹) ↾ ∅) = ∅
6		0ex 5257	. . . . . . 7 ⊢ ∅ ∈ V
7	5, 6	eqeltri 2824	. . . . . 6 ⊢ (recs(𝐹) ↾ ∅) ∈ V
8		0elon 6375	. . . . . . 7 ⊢ ∅ ∈ On
9	1	tfrlem15 8337	. . . . . . 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 231	. . . . 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 8335	. . . . 5 ⊢ ¬ recs(𝐹) ∈ V
15		simpr 484	. . . . . . . . . 10 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → dom recs(𝐹) = suc 𝑧)
16		df-suc 6326	. . . . . . . . . 10 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧})
17	15, 16	eqtrdi 2780	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → dom recs(𝐹) = (𝑧 ∪ {𝑧}))
18	17	reseq2d 5939	. . . . . . . 8 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → (recs(𝐹) ↾ dom recs(𝐹)) = (recs(𝐹) ↾ (𝑧 ∪ {𝑧})))
19	1	tfrlem6 8327	. . . . . . . . 9 ⊢ Rel recs(𝐹)
20		resdm 5986	. . . . . . . . 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 5953	. . . . . . . 8 ⊢ (recs(𝐹) ↾ (𝑧 ∪ {𝑧})) = ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧}))
23	18, 21, 22	3eqtr3g 2787	. . . . . . 7 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → recs(𝐹) = ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})))
24		vex 3448	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
25	24	sucid 6404	. . . . . . . . . 10 ⊢ 𝑧 ∈ suc 𝑧
26	25, 15	eleqtrrid 2835	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → 𝑧 ∈ dom recs(𝐹))
27	1	tfrlem9a 8331	. . . . . . . . 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 5386	. . . . . . . . 9 ⊢ {⟨𝑧, (recs(𝐹)‘𝑧)⟩} ∈ V
30	1	tfrlem7 8328	. . . . . . . . . 10 ⊢ Fun recs(𝐹)
31		funressn 7113	. . . . . . . . . 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 5272	. . . . . . . 8 ⊢ (recs(𝐹) ↾ {𝑧}) ∈ V
34		unexg 7699	. . . . . . . 8 ⊢ (((recs(𝐹) ↾ 𝑧) ∈ V ∧ (recs(𝐹) ↾ {𝑧}) ∈ V) → ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})) ∈ V)
35	28, 33, 34	sylancl 586	. . . . . . 7 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})) ∈ V)
36	23, 35	eqeltrd 2828	. . . . . 6 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → recs(𝐹) ∈ V)
37	36	rexlimiva 3126	. . . . 5 ⊢ (∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧 → recs(𝐹) ∈ V)
38	14, 37	mto 197	. . . 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 1430	. 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 206   ∧ wa 395   ∨ w3o 1085   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053  Vcvv 3444   ∪ cun 3909   ⊆ wss 3911  ∅c0 4292  {csn 4585  ⟨cop 4591  dom cdm 5631   ↾ cres 5633  Rel wrel 5636  Ord word 6319  Oncon0 6320  Lim wlim 6321  suc csuc 6322  Fun wfun 6493   Fn wfn 6494  ‘cfv 6499  recscrecs 8316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317

This theorem is referenced by:  tfr1a  8339