Theorem tfrlem9a 8317

Description: Lemma for transfinite recursion. Without using ax-rep 5224, 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 8314	. . . 4 ⊢ Fun recs(𝐹)
3		funfvop 6995	. . . 4 ⊢ ((Fun recs(𝐹) ∧ 𝐵 ∈ dom recs(𝐹)) → ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ recs(𝐹))
4	2, 3	mpan 690	. . 3 ⊢ (𝐵 ∈ dom recs(𝐹) → ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ recs(𝐹))
5	1	recsfval 8312	. . . . 5 ⊢ recs(𝐹) = ∪ 𝐴
6	5	eleq2i 2828	. . . 4 ⊢ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ recs(𝐹) ↔ ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ ∪ 𝐴)
7		eluni 4866	. . . 4 ⊢ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ ∪ 𝐴 ↔ ∃𝑔(⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴))
8	6, 7	bitri 275	. . 3 ⊢ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ recs(𝐹) ↔ ∃𝑔(⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴))
9	4, 8	sylib 218	. 2 ⊢ (𝐵 ∈ dom recs(𝐹) → ∃𝑔(⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴))
10		simprr 772	. . . 4 ⊢ ((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) → 𝑔 ∈ 𝐴)
11		vex 3444	. . . . 5 ⊢ 𝑔 ∈ V
12	1, 11	tfrlem3a 8308	. . . 4 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎 ∈ 𝑧 (𝑔‘𝑎) = (𝐹‘(𝑔 ↾ 𝑎))))
13	10, 12	sylib 218	. . 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 777	. . . . . . . . . 10 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝑔 ∈ 𝐴)
16		elssuni 4894	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝐴 → 𝑔 ⊆ ∪ 𝐴)
17	15, 16	syl 17	. . . . . . . . 9 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝑔 ⊆ ∪ 𝐴)
18	17, 5	sseqtrrdi 3975	. . . . . . . 8 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝑔 ⊆ recs(𝐹))
19		fndm 6595	. . . . . . . . . . . 12 ⊢ (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
20	19	ad2antll 729	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → dom 𝑔 = 𝑧)
21		simprl 770	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝑧 ∈ On)
22	20, 21	eqeltrd 2836	. . . . . . . . . 10 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → dom 𝑔 ∈ On)
23		eloni 6327	. . . . . . . . . 10 ⊢ (dom 𝑔 ∈ On → Ord dom 𝑔)
24	22, 23	syl 17	. . . . . . . . 9 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → Ord dom 𝑔)
25		simpll 766	. . . . . . . . . 10 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝐵 ∈ dom recs(𝐹))
26		fvexd 6849	. . . . . . . . . 10 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → (recs(𝐹)‘𝐵) ∈ V)
27		simplrl 776	. . . . . . . . . . 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 5099	. . . . . . . . . . 11 ⊢ (𝐵𝑔(recs(𝐹)‘𝐵) ↔ ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔)
29	27, 28	sylibr 234	. . . . . . . . . 10 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝐵𝑔(recs(𝐹)‘𝐵))
30		breldmg 5858	. . . . . . . . . 10 ⊢ ((𝐵 ∈ dom recs(𝐹) ∧ (recs(𝐹)‘𝐵) ∈ V ∧ 𝐵𝑔(recs(𝐹)‘𝐵)) → 𝐵 ∈ dom 𝑔)
31	25, 26, 29, 30	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝐵 ∈ dom 𝑔)
32		ordelss 6333	. . . . . . . . 9 ⊢ ((Ord dom 𝑔 ∧ 𝐵 ∈ dom 𝑔) → 𝐵 ⊆ dom 𝑔)
33	24, 31, 32	syl2anc 584	. . . . . . . 8 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝐵 ⊆ dom 𝑔)
34		fun2ssres 6537	. . . . . . . 8 ⊢ ((Fun recs(𝐹) ∧ 𝑔 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑔) → (recs(𝐹) ↾ 𝐵) = (𝑔 ↾ 𝐵))
35	14, 18, 33, 34	syl3anc 1373	. . . . . . 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 5988	. . . . . . . 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 2836	. . . . . 6 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → (recs(𝐹) ↾ 𝐵) ∈ V)
39	38	expr 456	. . . . 5 ⊢ (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) ∧ 𝑧 ∈ On) → (𝑔 Fn 𝑧 → (recs(𝐹) ↾ 𝐵) ∈ V))
40	39	adantrd 491	. . . 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 3137	. . 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 15	. 2 ⊢ ((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) → (recs(𝐹) ↾ 𝐵) ∈ V)
43	9, 42	exlimddv 1936	1 ⊢ (𝐵 ∈ dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2714  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ⊆ wss 3901  ⟨cop 4586  ∪ cuni 4863   class class class wbr 5098  dom cdm 5624   ↾ cres 5626  Ord word 6316  Oncon0 6317  Fun wfun 6486   Fn wfn 6487  ‘cfv 6492  recscrecs 8302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-ov 7361  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303

This theorem is referenced by:  tfrlem15  8323  tfrlem16  8324  rdgseg  8353