Theorem tfrlem16 8058

Description: Lemma for finite recursion. Without assuming ax-rep 5154, 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 8049	. . 3 ⊢ Ord dom recs(𝐹)
3		ordzsl 7579	. . 3 ⊢ (Ord dom recs(𝐹) ↔ (dom recs(𝐹) = ∅ ∨ ∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧 ∨ Lim dom recs(𝐹)))
4	2, 3	mpbi 233	. 2 ⊢ (dom recs(𝐹) = ∅ ∨ ∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧 ∨ Lim dom recs(𝐹))
5		res0 5829	. . . . . . 7 ⊢ (recs(𝐹) ↾ ∅) = ∅
6		0ex 5175	. . . . . . 7 ⊢ ∅ ∈ V
7	5, 6	eqeltri 2829	. . . . . 6 ⊢ (recs(𝐹) ↾ ∅) ∈ V
8		0elon 6225	. . . . . . 7 ⊢ ∅ ∈ On
9	1	tfrlem15 8057	. . . . . . 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 234	. . . . 5 ⊢ ∅ ∈ dom recs(𝐹)
12	11	n0ii 4225	. . . 4 ⊢ ¬ dom recs(𝐹) = ∅
13	12	pm2.21i 119	. . 3 ⊢ (dom recs(𝐹) = ∅ → Lim dom recs(𝐹))
14	1	tfrlem13 8055	. . . . 5 ⊢ ¬ recs(𝐹) ∈ V
15		simpr 488	. . . . . . . . . 10 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → dom recs(𝐹) = suc 𝑧)
16		df-suc 6178	. . . . . . . . . 10 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧})
17	15, 16	eqtrdi 2789	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → dom recs(𝐹) = (𝑧 ∪ {𝑧}))
18	17	reseq2d 5825	. . . . . . . 8 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → (recs(𝐹) ↾ dom recs(𝐹)) = (recs(𝐹) ↾ (𝑧 ∪ {𝑧})))
19	1	tfrlem6 8047	. . . . . . . . 9 ⊢ Rel recs(𝐹)
20		resdm 5870	. . . . . . . . 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 5839	. . . . . . . 8 ⊢ (recs(𝐹) ↾ (𝑧 ∪ {𝑧})) = ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧}))
23	18, 21, 22	3eqtr3g 2796	. . . . . . 7 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → recs(𝐹) = ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})))
24		vex 3402	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
25	24	sucid 6251	. . . . . . . . . 10 ⊢ 𝑧 ∈ suc 𝑧
26	25, 15	eleqtrrid 2840	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → 𝑧 ∈ dom recs(𝐹))
27	1	tfrlem9a 8051	. . . . . . . . 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 5298	. . . . . . . . 9 ⊢ {⟨𝑧, (recs(𝐹)‘𝑧)⟩} ∈ V
30	1	tfrlem7 8048	. . . . . . . . . 10 ⊢ Fun recs(𝐹)
31		funressn 6931	. . . . . . . . . 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 5190	. . . . . . . 8 ⊢ (recs(𝐹) ↾ {𝑧}) ∈ V
34		unexg 7490	. . . . . . . 8 ⊢ (((recs(𝐹) ↾ 𝑧) ∈ V ∧ (recs(𝐹) ↾ {𝑧}) ∈ V) → ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})) ∈ V)
35	28, 33, 34	sylancl 589	. . . . . . 7 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})) ∈ V)
36	23, 35	eqeltrd 2833	. . . . . 6 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → recs(𝐹) ∈ V)
37	36	rexlimiva 3191	. . . . 5 ⊢ (∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧 → recs(𝐹) ∈ V)
38	14, 37	mto 200	. . . 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 1428	. 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 209   ∧ wa 399   ∨ w3o 1087   = wceq 1542   ∈ wcel 2114  {cab 2716  ∀wral 3053  ∃wrex 3054  Vcvv 3398   ∪ cun 3841   ⊆ wss 3843  ∅c0 4211  {csn 4516  ⟨cop 4522  dom cdm 5525   ↾ cres 5527  Rel wrel 5530  Ord word 6171  Oncon0 6172  Lim wlim 6173  suc csuc 6174  Fun wfun 6333   Fn wfn 6334  ‘cfv 6339  recscrecs 8036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-wrecs 7976  df-recs 8037

This theorem is referenced by:  tfr1a  8059