Theorem tfrlem15 8382

Description: Lemma for transfinite recursion. Without assuming ax-rep 5232, we can show that all proper initial subsets of recs are sets, while nothing larger is a set. (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
tfrlem15	⊢ (𝐵 ∈ On → (𝐵 ∈ dom recs(𝐹) ↔ (recs(𝐹) ↾ 𝐵) ∈ V))

Distinct variable groups:   𝑥,𝑓,𝑦,𝐵   𝑓,𝐹,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥, 𝑦, 𝑓)

Proof of Theorem tfrlem15

Step	Hyp	Ref	Expression
1		tfrlem.1	. . . 4 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
2	1	tfrlem9a 8376	. . 3 ⊢ (𝐵 ∈ dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) ∈ V)
3	2	adantl 487	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ dom recs(𝐹)) → (recs(𝐹) ↾ 𝐵) ∈ V)
4	1	tfrlem13 8380	. . . 4 ⊢ ¬ recs(𝐹) ∈ V
5		simpr 490	. . . . 5 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → (recs(𝐹) ↾ 𝐵) ∈ V)
6		resss 5994	. . . . . . . 8 ⊢ (recs(𝐹) ↾ 𝐵) ⊆ recs(𝐹)
7	6	a1i 11	. . . . . . 7 ⊢ (dom recs(𝐹) ⊆ 𝐵 → (recs(𝐹) ↾ 𝐵) ⊆ recs(𝐹))
8	1	tfrlem6 8371	. . . . . . . . 9 ⊢ Rel recs(𝐹)
9		resdm 6019	. . . . . . . . 9 ⊢ (Rel recs(𝐹) → (recs(𝐹) ↾ dom recs(𝐹)) = recs(𝐹))
10	8, 9	ax-mp 5	. . . . . . . 8 ⊢ (recs(𝐹) ↾ dom recs(𝐹)) = recs(𝐹)
11		ssres2 5997	. . . . . . . 8 ⊢ (dom recs(𝐹) ⊆ 𝐵 → (recs(𝐹) ↾ dom recs(𝐹)) ⊆ (recs(𝐹) ↾ 𝐵))
12	10, 11	eqsstrrid 3970	. . . . . . 7 ⊢ (dom recs(𝐹) ⊆ 𝐵 → recs(𝐹) ⊆ (recs(𝐹) ↾ 𝐵))
13	7, 12	eqssd 3948	. . . . . 6 ⊢ (dom recs(𝐹) ⊆ 𝐵 → (recs(𝐹) ↾ 𝐵) = recs(𝐹))
14	13	eleq1d 2845	. . . . 5 ⊢ (dom recs(𝐹) ⊆ 𝐵 → ((recs(𝐹) ↾ 𝐵) ∈ V ↔ recs(𝐹) ∈ V))
15	5, 14	syl5ibcom 248	. . . 4 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → (dom recs(𝐹) ⊆ 𝐵 → recs(𝐹) ∈ V))
16	4, 15	mtoi 202	. . 3 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → ¬ dom recs(𝐹) ⊆ 𝐵)
17	1	tfrlem8 8374	. . . 4 ⊢ Ord dom recs(𝐹)
18		eloni 6367	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
19	18	adantr 486	. . . 4 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → Ord 𝐵)
20		ordtri1 6391	. . . . 5 ⊢ ((Ord dom recs(𝐹) ∧ Ord 𝐵) → (dom recs(𝐹) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ dom recs(𝐹)))
21	20	con2bid 357	. . . 4 ⊢ ((Ord dom recs(𝐹) ∧ Ord 𝐵) → (𝐵 ∈ dom recs(𝐹) ↔ ¬ dom recs(𝐹) ⊆ 𝐵))
22	17, 19, 21	sylancr 599	. . 3 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → (𝐵 ∈ dom recs(𝐹) ↔ ¬ dom recs(𝐹) ⊆ 𝐵))
23	16, 22	mpbird 260	. 2 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → 𝐵 ∈ dom recs(𝐹))
24	3, 23	impbida 813	1 ⊢ (𝐵 ∈ On → (𝐵 ∈ dom recs(𝐹) ↔ (recs(𝐹) ↾ 𝐵) ∈ V))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3899  dom cdm 5655   ↾ cres 5657  Rel wrel 5660  Ord word 6356  Oncon0 6357   Fn wfn 6528  ‘cfv 6533  recscrecs 8360

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fo 6539  df-fv 6541  df-ov 7417  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361

This theorem is used by:  tfrlem16  8383  tfr2b  8386