Theorem tfrlem9a 8432

Description: Lemma for transfinite recursion. Without using ax-rep 5238, show that all the restrictions of recs are sets. (Contributed by Mario Carneiro, 16-Nov-2014.)

Hypothesis

Ref	Expression
tfrlem.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

Assertion

Ref	Expression
tfrlem9a	⊢ (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 tfrlem9a

Dummy variables 𝑔 𝑧 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfrlem.1	. . . . 5 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
2	1	tfrlem7 8429	. . . 4 ⊢ Fun recs(𝐹)
3		funfvop 7080	. . . 4 ⊢ ((Fun recs(𝐹) ∧ 𝐵 ∈ dom recs(𝐹)) → ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ recs(𝐹))
4	2, 3	mpan 703	. . 3 ⊢ (𝐵 ∈ dom recs(𝐹) → ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ recs(𝐹))
5	1	recsfval 8426	. . . . 5 ⊢ recs(𝐹) = ∪ 𝐴
6	5	eleq2i 2856	. . . 4 ⊢ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ recs(𝐹) ↔ ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ ∪ 𝐴)
7		eluni 4875	. . . 4 ⊢ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ ∪ 𝐴 ↔ ∃𝑔(⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴))
8	6, 7	bitri 278	. . 3 ⊢ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ recs(𝐹) ↔ ∃𝑔(⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴))
9	4, 8	sylib 221	. 2 ⊢ (𝐵 ∈ dom recs(𝐹) → ∃𝑔(⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴))
10		simprr 785	. . . 4 ⊢ ((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) → 𝑔 ∈ 𝐴)
11		vex 3458	. . . . 5 ⊢ 𝑔 ∈ V
12	1, 11	tfrlem3a 8422	. . . 4 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))))
13	10, 12	sylib 221	. . 3 ⊢ ((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) → ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))))
14	2	a1i 11	. . . . . . . 8 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → Fun recs(𝐹))
15		simplrr 790	. . . . . . . . . 10 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝑔 ∈ 𝐴)
16		elssuni 4904	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝐴 → 𝑔 ⊆ ∪ 𝐴)
17	15, 16	syl 18	. . . . . . . . 9 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝑔 ⊆ ∪ 𝐴)
18	17, 5	sseqtrrdi 3954	. . . . . . . 8 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝑔 ⊆ recs(𝐹))
19		fndm 6672	. . . . . . . . . . . 12 ⊢ (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
20	19	ad2antll 742	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → dom 𝑔 = 𝑧)
21		simprl 783	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝑧 ∈ On)
22	20, 21	eqeltrd 2864	. . . . . . . . . 10 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → dom 𝑔 ∈ On)
23		eloni 6398	. . . . . . . . . 10 ⊢ (dom 𝑔 ∈ On → Ord dom 𝑔)
24	22, 23	syl 18	. . . . . . . . 9 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → Ord dom 𝑔)
25		simpll 779	. . . . . . . . . 10 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝐵 ∈ dom recs(𝐹))
26		fvexd 6931	. . . . . . . . . 10 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → (recs(𝐹)‘𝐵) ∈ V)
27		simplrl 789	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔)
28		df-br 5109	. . . . . . . . . . 11 ⊢ (𝐵𝑔(recs(𝐹)‘𝐵) ↔ ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔)
29	27, 28	sylibr 237	. . . . . . . . . 10 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝐵𝑔(recs(𝐹)‘𝐵))
30		breldmg 5903	. . . . . . . . . 10 ⊢ ((𝐵 ∈ dom recs(𝐹) ∧ (recs(𝐹)‘𝐵) ∈ V ∧ 𝐵𝑔(recs(𝐹)‘𝐵)) → 𝐵 ∈ dom 𝑔)
31	25, 26, 29, 30	syl3anc 1399	. . . . . . . . 9 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝐵 ∈ dom 𝑔)
32		ordelss 6404	. . . . . . . . 9 ⊢ ((Ord dom 𝑔 ∧ 𝐵 ∈ dom 𝑔) → 𝐵 ⊆ dom 𝑔)
33	24, 31, 32	syl2anc 596	. . . . . . . 8 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝐵 ⊆ dom 𝑔)
34		fun2ssres 6613	. . . . . . . 8 ⊢ ((Fun recs(𝐹) ∧ 𝑔 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑔) → (recs(𝐹) ↾ 𝐵) = (𝑔 ↾ 𝐵))
35	14, 18, 33, 34	syl3anc 1399	. . . . . . 7 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → (recs(𝐹) ↾ 𝐵) = (𝑔 ↾ 𝐵))
36	11	resex 6036	. . . . . . . 8 ⊢ (𝑔 ↾ 𝐵) ∈ V
37	36	a1i 11	. . . . . . 7 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → (𝑔 ↾ 𝐵) ∈ V)
38	35, 37	eqeltrd 2864	. . . . . 6 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → (recs(𝐹) ↾ 𝐵) ∈ V)
39	38	expr 462	. . . . 5 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ 𝑧 ∈ On) → (𝑔 Fn 𝑧 → (recs(𝐹) ↾ 𝐵) ∈ V))
40	39	adantrd 497	. . . 4 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ 𝑧 ∈ On) → ((𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) → (recs(𝐹) ↾ 𝐵) ∈ V))
41	40	rexlimdva 3167	. . 3 ⊢ ((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) → (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))) → (recs(𝐹) ↾ 𝐵) ∈ V))
42	13, 41	mpd 16	. 2 ⊢ ((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) → (recs(𝐹) ↾ 𝐵) ∈ V)
43	9, 42	exlimddv 1970	1 ⊢ (𝐵 ∈ dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  ⟨cop 4840  ∪ cuni 4872   class class class wbr 5108  dom cdm 5658   ↾ cres 5660  Ord word 6387  Oncon0 6388  Fun wfun 6560   Fn wfn 6561  ‘cfv 6566  recscrecs 8416

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fo 6572  df-fv 6574  df-ov 7454  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417

This theorem is used by:  tfrlem15  8438  tfrlem16  8439  rdgseg  8468