Theorem tfrlem9a 8369

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 8366	. . . 4 ⊢ Fun recs(𝐹)
3		funfvop 7045	. . . 4 ⊢ ((Fun recs(𝐹) ∧ 𝐵 ∈ dom recs(𝐹)) → ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ recs(𝐹))
4	2, 3	mpan 702	. . 3 ⊢ (𝐵 ∈ dom recs(𝐹) → ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ recs(𝐹))
5	1	recsfval 8363	. . . . 5 ⊢ recs(𝐹) = ∪ 𝐴
6	5	eleq2i 2855	. . . 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 784	. . . 4 ⊢ ((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) → 𝑔 ∈ 𝐴)
11		vex 3459	. . . . 5 ⊢ 𝑔 ∈ V
12	1, 11	tfrlem3a 8359	. . . 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 789	. . . . . . . . . 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 3978	. . . . . . . 8 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝑔 ⊆ recs(𝐹))
19		fndm 6638	. . . . . . . . . . . 12 ⊢ (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
20	19	ad2antll 741	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → dom 𝑔 = 𝑧)
21		simprl 782	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝑧 ∈ On)
22	20, 21	eqeltrd 2863	. . . . . . . . . 10 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → dom 𝑔 ∈ On)
23		eloni 6370	. . . . . . . . . 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 778	. . . . . . . . . 10 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝐵 ∈ dom recs(𝐹))
26		fvexd 6896	. . . . . . . . . 10 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → (recs(𝐹)‘𝐵) ∈ V)
27		simplrl 788	. . . . . . . . . . 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 5110	. . . . . . . . . . 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 5899	. . . . . . . . . 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 6376	. . . . . . . . 9 ⊢ ((Ord dom 𝑔 ∧ 𝐵 ∈ dom 𝑔) → 𝐵 ⊆ dom 𝑔)
33	24, 31, 32	syl2anc 595	. . . . . . . 8 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝐵 ⊆ dom 𝑔)
34		fun2ssres 6581	. . . . . . . 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 6028	. . . . . . . 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 2863	. . . . . 6 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → (recs(𝐹) ↾ 𝐵) ∈ V)
39	38	expr 461	. . . . 5 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ 𝑧 ∈ On) → (𝑔 Fn 𝑧 → (recs(𝐹) ↾ 𝐵) ∈ V))
40	39	adantrd 496	. . . 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 3166	. . 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 1965	1 ⊢ (𝐵 ∈ dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ⊆ wss 3905  ⟨cop 4595  ∪ cuni 4872   class class class wbr 5109  dom cdm 5661   ↾ cres 5663  Ord word 6359  Oncon0 6360  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536  recscrecs 8353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-ov 7413  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354

This theorem is referenced by:  tfrlem15  8375  tfrlem16  8376  rdgseg  8405