Theorem tfrlem11 8371

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 6430	. 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 8370	. . . . . . . 8 ⊢ (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
5		fnfun 6635	. . . . . . . 8 ⊢ (𝐶 Fn suc dom recs(𝐹) → Fun 𝐶)
6	4, 5	syl 18	. . . . . . 7 ⊢ (dom recs(𝐹) ∈ On → Fun 𝐶)
7		ssun1 4131	. . . . . . . . 9 ⊢ recs(𝐹) ⊆ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
8	7, 3	sseqtrri 3986	. . . . . . . 8 ⊢ recs(𝐹) ⊆ 𝐶
9	2	tfrlem9 8368	. . . . . . . . 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 1136	. . . . . . . . . . 11 ⊢ (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ∈ dom recs(𝐹)) → (𝐶‘𝐵) = (recs(𝐹)‘𝐵))
12	11	adantrl 728	. . . . . . . . . 10 ⊢ (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐶‘𝐵) = (recs(𝐹)‘𝐵))
13		onelss 6403	. . . . . . . . . . . 12 ⊢ (dom recs(𝐹) ∈ On → (𝐵 ∈ dom recs(𝐹) → 𝐵 ⊆ dom recs(𝐹)))
14	13	imp 411	. . . . . . . . . . 11 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹)) → 𝐵 ⊆ dom recs(𝐹))
15		fun2ssres 6581	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶 ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐶 ↾ 𝐵) = (recs(𝐹) ↾ 𝐵))
16	15	3expa 1136	. . . . . . . . . . . 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 604	. . . . . . . . . 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 2779	. . . . . . . . 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 249	. . . . . . . 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 713	. . . . . . 7 ⊢ ((Fun 𝐶 ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐵 ∈ dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))
22	6, 21	sylan 591	. . . . . 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 425	. . . . 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 53	. . . 4 ⊢ (dom recs(𝐹) ∈ On → (𝐵 ∈ dom recs(𝐹) → (𝐵 ∈ dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵)))))
25	24	pm2.43d 54	. . 3 ⊢ (dom recs(𝐹) ∈ On → (𝐵 ∈ dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))
26		opex 5445	. . . . . . . . 9 ⊢ ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ V
27	26	snid 4628	. . . . . . . 8 ⊢ ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ {⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩}
28		opeq1 4838	. . . . . . . . . . 11 ⊢ (𝐵 = dom recs(𝐹) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘(𝐶 ↾ 𝐵))⟩)
29	28	adantl 486	. . . . . . . . . 10 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘(𝐶 ↾ 𝐵))⟩)
30		eqimss 3995	. . . . . . . . . . . . . 14 ⊢ (𝐵 = dom recs(𝐹) → 𝐵 ⊆ dom recs(𝐹))
31	8, 15	mp3an2 1478	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐶 ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐶 ↾ 𝐵) = (recs(𝐹) ↾ 𝐵))
32	6, 30, 31	syl2an 607	. . . . . . . . . . . . 13 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶 ↾ 𝐵) = (recs(𝐹) ↾ 𝐵))
33		reseq2 5973	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) = (recs(𝐹) ↾ dom recs(𝐹)))
34	2	tfrlem6 8364	. . . . . . . . . . . . . . . 16 ⊢ Rel recs(𝐹)
35		resdm 6025	. . . . . . . . . . . . . . . 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 2814	. . . . . . . . . . . . . 14 ⊢ (𝐵 = dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) = recs(𝐹))
38	37	adantl 486	. . . . . . . . . . . . 13 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (recs(𝐹) ↾ 𝐵) = recs(𝐹))
39	32, 38	eqtrd 2798	. . . . . . . . . . . 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 4845	. . . . . . . . . 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 2798	. . . . . . . . 9 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩)
43	42	sneqd 4601	. . . . . . . 8 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → {⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩} = {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
44	27, 43	eleqtrid 2869	. . . . . . 7 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
45		elun2 4136	. . . . . . 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 18	. . . . . 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 2874	. . . . 5 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ 𝐶)
48		simpr 489	. . . . . . 7 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐵 = dom recs(𝐹))
49		sucidg 6444	. . . . . . . 8 ⊢ (dom recs(𝐹) ∈ On → dom recs(𝐹) ∈ suc dom recs(𝐹))
50	49	adantr 485	. . . . . . 7 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → dom recs(𝐹) ∈ suc dom recs(𝐹))
51	48, 50	eqeltrd 2863	. . . . . 6 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐵 ∈ suc dom recs(𝐹))
52		fnopfvb 6932	. . . . . 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 697	. . . . 5 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ((𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵)) ↔ ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ 𝐶))
54	47, 53	mpbird 260	. . . 4 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵)))
55	54	ex 417	. . 3 ⊢ (dom recs(𝐹) ∈ On → (𝐵 = dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))
56	25, 55	jaod 872	. 2 ⊢ (dom recs(𝐹) ∈ On → ((𝐵 ∈ dom recs(𝐹) ∨ 𝐵 = dom recs(𝐹)) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))
57	1, 56	syl5 35	1 ⊢ (dom recs(𝐹) ∈ On → (𝐵 ∈ suc dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143  {cab 2741  ∀wral 3079  ∃wrex 3089   ∪ cun 3903   ⊆ wss 3905  {csn 4589  ⟨cop 4595  dom cdm 5661   ↾ cres 5663  Rel wrel 5666  Oncon0 6360  suc csuc 6362  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-suc 6366  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:  tfrlem12  8372