Theorem tfrlem16 8364

Description: Lemma for finite recursion. Without assuming ax-rep 5227, 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 8355	. . 3 ⊢ Ord dom recs(𝐹)
3		ordzsl 7825	. . 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 5969	. . . . . . 7 ⊢ (recs(𝐹) ↾ ∅) = ∅
6		0ex 5257	. . . . . . 7 ⊢ ∅ ∈ V
7	5, 6	eqeltri 2858	. . . . . 6 ⊢ (recs(𝐹) ↾ ∅) ∈ V
8		0elon 6401	. . . . . . 7 ⊢ ∅ ∈ On
9	1	tfrlem15 8363	. . . . . . 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 4295	. . . 4 ⊢ ¬ dom recs(𝐹) = ∅
13	12	pm2.21i 119	. . 3 ⊢ (dom recs(𝐹) = ∅ → Lim dom recs(𝐹))
14	1	tfrlem13 8361	. . . . 5 ⊢ ¬ recs(𝐹) ∈ V
15		simpr 488	. . . . . . . . . 10 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → dom recs(𝐹) = suc 𝑧)
16		df-suc 6352	. . . . . . . . . 10 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧})
17	15, 16	eqtrdi 2813	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → dom recs(𝐹) = (𝑧 ∪ {𝑧}))
18	17	reseq2d 5965	. . . . . . . 8 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → (recs(𝐹) ↾ dom recs(𝐹)) = (recs(𝐹) ↾ (𝑧 ∪ {𝑧})))
19	1	tfrlem6 8352	. . . . . . . . 9 ⊢ Rel recs(𝐹)
20		resdm 6012	. . . . . . . . 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 5979	. . . . . . . 8 ⊢ (recs(𝐹) ↾ (𝑧 ∪ {𝑧})) = ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧}))
23	18, 21, 22	3eqtr3g 2820	. . . . . . 7 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → recs(𝐹) = ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})))
24		vex 3458	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
25	24	sucid 6430	. . . . . . . . . 10 ⊢ 𝑧 ∈ suc 𝑧
26	25, 15	eleqtrrid 2869	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → 𝑧 ∈ dom recs(𝐹))
27	1	tfrlem9a 8357	. . . . . . . . 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 5396	. . . . . . . . 9 ⊢ {⟨𝑧, (recs(𝐹)‘𝑧)⟩} ∈ V
30	1	tfrlem7 8354	. . . . . . . . . 10 ⊢ Fun recs(𝐹)
31		funressn 7142	. . . . . . . . . 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 5278	. . . . . . . 8 ⊢ (recs(𝐹) ↾ {𝑧}) ∈ V
34		unexg 7726	. . . . . . . 8 ⊢ (((recs(𝐹) ↾ 𝑧) ∈ V ∧ (recs(𝐹) ↾ {𝑧}) ∈ V) → ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})) ∈ V)
35	28, 33, 34	sylancl 595	. . . . . . 7 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})) ∈ V)
36	23, 35	eqeltrd 2862	. . . . . 6 ⊢ ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → recs(𝐹) ∈ V)
37	36	rexlimiva 3155	. . . . 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 1447	. 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 399   ∨ w3o 1097   = wceq 1560   ∈ wcel 2142  {cab 2740  ∀wral 3076  ∃wrex 3086  Vcvv 3454   ∪ cun 3902   ⊆ wss 3904  ∅c0 4285  {csn 4582  ⟨cop 4588  dom cdm 5647   ↾ cres 5649  Rel wrel 5652  Ord word 6345  Oncon0 6346  Lim wlim 6347  suc csuc 6348  Fun wfun 6515   Fn wfn 6516  ‘cfv 6521  recscrecs 8341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342

This theorem is referenced by:  tfr1a  8365