Theorem tfrlem16 8361

Description: Lemma for finite recursion. Without assuming ax-rep 5234, 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 8352	. . 3 ⊢ Ord dom recs(𝐹)
3		ordzsl 7821	. . 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 5954	. . . . . . 7 ⊢ (recs(𝐹) ↾ ∅) = ∅
6		0ex 5262	. . . . . . 7 ⊢ ∅ ∈ V
7	5, 6	eqeltri 2824	. . . . . 6 ⊢ (recs(𝐹) ↾ ∅) ∈ V
8		0elon 6387	. . . . . . 7 ⊢ ∅ ∈ On
9	1	tfrlem15 8360	. . . . . . 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 4306	. . . 4 ⊢ ¬ dom recs(𝐹) = ∅
13	12	pm2.21i 119	. . 3 ⊢ (dom recs(𝐹) = ∅ → Lim dom recs(𝐹))
14	1	tfrlem13 8358	. . . . 5 ⊢ ¬ recs(𝐹) ∈ V
15		simpr 484	. . . . . . . . . 10 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → dom recs(𝐹) = suc 𝑧)
16		df-suc 6338	. . . . . . . . . 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 5950	. . . . . . . 8 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → (recs(𝐹) ↾ dom recs(𝐹)) = (recs(𝐹) ↾ (𝑧 ∪ {𝑧})))
19	1	tfrlem6 8350	. . . . . . . . 9 ⊢ Rel recs(𝐹)
20		resdm 5997	. . . . . . . . 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 5964	. . . . . . . 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 3451	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
25	24	sucid 6416	. . . . . . . . . 10 ⊢ 𝑧 ∈ suc 𝑧
26	25, 15	eleqtrrid 2835	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → 𝑧 ∈ dom recs(𝐹))
27	1	tfrlem9a 8354	. . . . . . . . 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 5391	. . . . . . . . 9 ⊢ {⟨𝑧, (recs(𝐹)‘𝑧)⟩} ∈ V
30	1	tfrlem7 8351	. . . . . . . . . 10 ⊢ Fun recs(𝐹)
31		funressn 7131	. . . . . . . . . 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 5277	. . . . . . . 8 ⊢ (recs(𝐹) ↾ {𝑧}) ∈ V
34		unexg 7719	. . . . . . . 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 3447   ∪ cun 3912   ⊆ wss 3914  ∅c0 4296  {csn 4589  ⟨cop 4595  dom cdm 5638   ↾ cres 5640  Rel wrel 5643  Ord word 6331  Oncon0 6332  Lim wlim 6333  suc csuc 6334  Fun wfun 6505   Fn wfn 6506  ‘cfv 6511  recscrecs 8339

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 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

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 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340

This theorem is referenced by:  tfr1a  8362