Theorem tfrlem11 8375

Description: Lemma for transfinite recursion. Compute the value of 𝐶. (Contributed by NM, 18-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)

Hypotheses

Ref	Expression
tfrlem.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
tfrlem.3	⊢ 𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})

Assertion

Ref	Expression
tfrlem11	⊢ (dom recs(𝐹) ∈ On → (𝐵 ∈ suc dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))

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

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

Proof of Theorem tfrlem11

Step	Hyp	Ref	Expression
1		elsuci 6423	. 2 ⊢ (𝐵 ∈ suc dom recs(𝐹) → (𝐵 ∈ dom recs(𝐹) ∨ 𝐵 = dom recs(𝐹)))
2		tfrlem.1	. . . . . . . . 9 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
3		tfrlem.3	. . . . . . . . 9 ⊢ 𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
4	2, 3	tfrlem10 8374	. . . . . . . 8 ⊢ (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
5		fnfun 6641	. . . . . . . 8 ⊢ (𝐶 Fn suc dom recs(𝐹) → Fun 𝐶)
6	4, 5	syl 17	. . . . . . 7 ⊢ (dom recs(𝐹) ∈ On → Fun 𝐶)
7		ssun1 4170	. . . . . . . . 9 ⊢ recs(𝐹) ⊆ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
8	7, 3	sseqtrri 4017	. . . . . . . 8 ⊢ recs(𝐹) ⊆ 𝐶
9	2	tfrlem9 8372	. . . . . . . . 9 ⊢ (𝐵 ∈ dom recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
10		funssfv 6902	. . . . . . . . . . . 12 ⊢ ((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶 ∧ 𝐵 ∈ dom recs(𝐹)) → (𝐶‘𝐵) = (recs(𝐹)‘𝐵))
11	10	3expa 1119	. . . . . . . . . . 11 ⊢ (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ∈ dom recs(𝐹)) → (𝐶‘𝐵) = (recs(𝐹)‘𝐵))
12	11	adantrl 715	. . . . . . . . . 10 ⊢ (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐶‘𝐵) = (recs(𝐹)‘𝐵))
13		onelss 6398	. . . . . . . . . . . 12 ⊢ (dom recs(𝐹) ∈ On → (𝐵 ∈ dom recs(𝐹) → 𝐵 ⊆ dom recs(𝐹)))
14	13	imp 408	. . . . . . . . . . 11 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹)) → 𝐵 ⊆ dom recs(𝐹))
15		fun2ssres 6585	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶 ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐶 ↾ 𝐵) = (recs(𝐹) ↾ 𝐵))
16	15	3expa 1119	. . . . . . . . . . . 12 ⊢ (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐶 ↾ 𝐵) = (recs(𝐹) ↾ 𝐵))
17	16	fveq2d 6885	. . . . . . . . . . 11 ⊢ (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐹‘(𝐶 ↾ 𝐵)) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
18	14, 17	sylan2 594	. . . . . . . . . 10 ⊢ (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐹‘(𝐶 ↾ 𝐵)) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
19	12, 18	eqeq12d 2749	. . . . . . . . 9 ⊢ (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → ((𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵)) ↔ (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))
20	9, 19	imbitrrid 245	. . . . . . . 8 ⊢ (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐵 ∈ dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))
21	8, 20	mpanl2 700	. . . . . . 7 ⊢ ((Fun 𝐶 ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐵 ∈ dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))
22	6, 21	sylan 581	. . . . . 6 ⊢ ((dom recs(𝐹) ∈ On ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐵 ∈ dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))
23	22	exp32 422	. . . . 5 ⊢ (dom recs(𝐹) ∈ On → (dom recs(𝐹) ∈ On → (𝐵 ∈ dom recs(𝐹) → (𝐵 ∈ dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))))
24	23	pm2.43i 52	. . . 4 ⊢ (dom recs(𝐹) ∈ On → (𝐵 ∈ dom recs(𝐹) → (𝐵 ∈ dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵)))))
25	24	pm2.43d 53	. . 3 ⊢ (dom recs(𝐹) ∈ On → (𝐵 ∈ dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))
26		opex 5460	. . . . . . . . 9 ⊢ ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ V
27	26	snid 4660	. . . . . . . 8 ⊢ ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ {⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩}
28		opeq1 4869	. . . . . . . . . . 11 ⊢ (𝐵 = dom recs(𝐹) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘(𝐶 ↾ 𝐵))⟩)
29	28	adantl 483	. . . . . . . . . 10 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘(𝐶 ↾ 𝐵))⟩)
30		eqimss 4038	. . . . . . . . . . . . . 14 ⊢ (𝐵 = dom recs(𝐹) → 𝐵 ⊆ dom recs(𝐹))
31	8, 15	mp3an2 1450	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐶 ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐶 ↾ 𝐵) = (recs(𝐹) ↾ 𝐵))
32	6, 30, 31	syl2an 597	. . . . . . . . . . . . 13 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶 ↾ 𝐵) = (recs(𝐹) ↾ 𝐵))
33		reseq2 5971	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) = (recs(𝐹) ↾ dom recs(𝐹)))
34	2	tfrlem6 8369	. . . . . . . . . . . . . . . 16 ⊢ Rel recs(𝐹)
35		resdm 6021	. . . . . . . . . . . . . . . 16 ⊢ (Rel recs(𝐹) → (recs(𝐹) ↾ dom recs(𝐹)) = recs(𝐹))
36	34, 35	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (recs(𝐹) ↾ dom recs(𝐹)) = recs(𝐹)
37	33, 36	eqtrdi 2789	. . . . . . . . . . . . . 14 ⊢ (𝐵 = dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) = recs(𝐹))
38	37	adantl 483	. . . . . . . . . . . . 13 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (recs(𝐹) ↾ 𝐵) = recs(𝐹))
39	32, 38	eqtrd 2773	. . . . . . . . . . . 12 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶 ↾ 𝐵) = recs(𝐹))
40	39	fveq2d 6885	. . . . . . . . . . 11 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐹‘(𝐶 ↾ 𝐵)) = (𝐹‘recs(𝐹)))
41	40	opeq2d 4876	. . . . . . . . . 10 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨dom recs(𝐹), (𝐹‘(𝐶 ↾ 𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩)
42	29, 41	eqtrd 2773	. . . . . . . . 9 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩)
43	42	sneqd 4636	. . . . . . . 8 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → {⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩} = {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
44	27, 43	eleqtrid 2840	. . . . . . 7 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
45		elun2 4175	. . . . . . 7 ⊢ (⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}))
46	44, 45	syl 17	. . . . . 6 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}))
47	46, 3	eleqtrrdi 2845	. . . . 5 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ 𝐶)
48		simpr 486	. . . . . . 7 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐵 = dom recs(𝐹))
49		sucidg 6437	. . . . . . . 8 ⊢ (dom recs(𝐹) ∈ On → dom recs(𝐹) ∈ suc dom recs(𝐹))
50	49	adantr 482	. . . . . . 7 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → dom recs(𝐹) ∈ suc dom recs(𝐹))
51	48, 50	eqeltrd 2834	. . . . . 6 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐵 ∈ suc dom recs(𝐹))
52		fnopfvb 6935	. . . . . 6 ⊢ ((𝐶 Fn suc dom recs(𝐹) ∧ 𝐵 ∈ suc dom recs(𝐹)) → ((𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵)) ↔ ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ 𝐶))
53	4, 51, 52	syl2an2r 684	. . . . 5 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ((𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵)) ↔ ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ 𝐶))
54	47, 53	mpbird 257	. . . 4 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵)))
55	54	ex 414	. . 3 ⊢ (dom recs(𝐹) ∈ On → (𝐵 = dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))
56	25, 55	jaod 858	. 2 ⊢ (dom recs(𝐹) ∈ On → ((𝐵 ∈ dom recs(𝐹) ∨ 𝐵 = dom recs(𝐹)) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))
57	1, 56	syl5 34	1 ⊢ (dom recs(𝐹) ∈ On → (𝐵 ∈ suc dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  {cab 2710  ∀wral 3062  ∃wrex 3071   ∪ cun 3944   ⊆ wss 3946  {csn 4624  ⟨cop 4630  dom cdm 5672   ↾ cres 5674  Rel wrel 5677  Oncon0 6356  suc csuc 6358  Fun wfun 6529   Fn wfn 6530  ‘cfv 6535  recscrecs 8357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423  ax-un 7712

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6292  df-ord 6359  df-on 6360  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-fo 6541  df-fv 6543  df-ov 7399  df-2nd 7963  df-frecs 8253  df-wrecs 8284  df-recs 8358

This theorem is referenced by:  tfrlem12  8376