Theorem tfrlem9a 8401

Description: Lemma for transfinite recursion. Without using ax-rep 5231, 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 8398	. . . 4 ⊢ Fun recs(𝐹)
3		funfvop 7054	. . . 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 8395	. . . . 5 ⊢ recs(𝐹) = ∪ 𝐴
6	5	eleq2i 2852	. . . 4 ⊢ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ recs(𝐹) ↔ ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ ∪ 𝐴)
7		eluni 4869	. . . 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 3454	. . . . 5 ⊢ 𝑔 ∈ V
12	1, 11	tfrlem3a 8391	. . . 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 4898	. . . . . . . . . 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 3950	. . . . . . . 8 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝑔 ⊆ recs(𝐹))
19		fndm 6647	. . . . . . . . . . . 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 2860	. . . . . . . . . 10 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → dom 𝑔 ∈ On)
23		eloni 6375	. . . . . . . . . 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 6905	. . . . . . . . . 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 5103	. . . . . . . . . . 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 5892	. . . . . . . . . 10 ⊢ ((𝐵 ∈ dom recs(𝐹) ∧ (recs(𝐹)‘𝐵) ∈ V ∧ 𝐵𝑔(recs(𝐹)‘𝐵)) → 𝐵 ∈ dom 𝑔)
31	25, 26, 29, 30	syl3anc 1398	. . . . . . . . 9 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝐵 ∈ dom 𝑔)
32		ordelss 6381	. . . . . . . . 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 6590	. . . . . . . 8 ⊢ ((Fun recs(𝐹) ∧ 𝑔 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑔) → (recs(𝐹) ↾ 𝐵) = (𝑔 ↾ 𝐵))
35	14, 18, 33, 34	syl3anc 1398	. . . . . . 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 6020	. . . . . . . 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 2860	. . . . . 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 3163	. . 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 1967	1 ⊢ (𝐵 ∈ dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3896  ⟨cop 4834  ∪ cuni 4866   class class class wbr 5102  dom cdm 5651   ↾ cres 5653  Ord word 6364  Oncon0 6365  Fun wfun 6537   Fn wfn 6538  ‘cfv 6543  recscrecs 8385

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-ov 7427  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386

This theorem is used by:  tfrlem15  8407  tfrlem16  8408  rdgseg  8437