Theorem cfsmolem 10227

Description: Lemma for cfsmo 10228. (Contributed by Mario Carneiro, 28-Feb-2013.)

Hypotheses

Ref	Expression
cfsmolem.2	⊢ 𝐹 = (𝑧 ∈ V ↦ ((𝑔‘dom 𝑧) ∪ ∪ 𝑡 ∈ dom 𝑧 suc (𝑧‘𝑡)))
cfsmolem.3	⊢ 𝐺 = (recs(𝐹) ↾ (cf‘𝐴))

Assertion

Ref	Expression
cfsmolem	⊢ (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))

Distinct variable groups:   𝑓,𝑔,𝑡,𝑤,𝑧,𝐴   𝑓,𝐹,𝑡,𝑧   𝑓,𝐺,𝑤,𝑧

Allowed substitution hints:   𝐹(𝑤,𝑔)   𝐺(𝑡,𝑔)

Proof of Theorem cfsmolem

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cff1 10215	. 2 ⊢ (𝐴 ∈ On → ∃𝑔(𝑔:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔‘𝑤)))
2		cfon 10211	. . . . . . . . . . . 12 ⊢ (cf‘𝐴) ∈ On
3	2	oneli 6461	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (cf‘𝐴) → 𝑥 ∈ On)
4	3	3ad2ant3 1148	. . . . . . . . . 10 ⊢ ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → 𝑥 ∈ On)
5		eleq1w 2845	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (cf‘𝐴) ↔ 𝑦 ∈ (cf‘𝐴)))
6	5	3anbi3d 1463	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ↔ (𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On ∧ 𝑦 ∈ (cf‘𝐴))))
7		fveq2 6867	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝐺‘𝑥) = (𝐺‘𝑦))
8	7	eleq1d 2847	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝐺‘𝑥) ∈ 𝐴 ↔ (𝐺‘𝑦) ∈ 𝐴))
9	6, 8	imbi12d 346	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (𝐺‘𝑥) ∈ 𝐴) ↔ ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On ∧ 𝑦 ∈ (cf‘𝐴)) → (𝐺‘𝑦) ∈ 𝐴)))
10		simpl1 1205	. . . . . . . . . . . . . . 15 ⊢ (((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦 ∈ 𝑥) → 𝑔:(cf‘𝐴)–1-1→𝐴)
11		simpl2 1206	. . . . . . . . . . . . . . 15 ⊢ (((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦 ∈ 𝑥) → 𝐴 ∈ On)
12		ontr1 6393	. . . . . . . . . . . . . . . . . 18 ⊢ ((cf‘𝐴) ∈ On → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (cf‘𝐴)) → 𝑦 ∈ (cf‘𝐴)))
13	2, 12	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (cf‘𝐴)) → 𝑦 ∈ (cf‘𝐴))
14	13	ancoms 462	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (cf‘𝐴) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (cf‘𝐴))
15	14	3ad2antl3 1201	. . . . . . . . . . . . . . 15 ⊢ (((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (cf‘𝐴))
16		pm2.27 42	. . . . . . . . . . . . . . 15 ⊢ ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On ∧ 𝑦 ∈ (cf‘𝐴)) → (((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On ∧ 𝑦 ∈ (cf‘𝐴)) → (𝐺‘𝑦) ∈ 𝐴) → (𝐺‘𝑦) ∈ 𝐴))
17	10, 11, 15, 16	syl3anc 1390	. . . . . . . . . . . . . 14 ⊢ (((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦 ∈ 𝑥) → (((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On ∧ 𝑦 ∈ (cf‘𝐴)) → (𝐺‘𝑦) ∈ 𝐴) → (𝐺‘𝑦) ∈ 𝐴))
18	17	ralimdva 3174	. . . . . . . . . . . . 13 ⊢ ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (∀𝑦 ∈ 𝑥 ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On ∧ 𝑦 ∈ (cf‘𝐴)) → (𝐺‘𝑦) ∈ 𝐴) → ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴))
19		cfsmolem.3	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐺 = (recs(𝐹) ↾ (cf‘𝐴))
20	19	fveq1i 6868	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺‘𝑥) = ((recs(𝐹) ↾ (cf‘𝐴))‘𝑥)
21		fvres 6886	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (cf‘𝐴) → ((recs(𝐹) ↾ (cf‘𝐴))‘𝑥) = (recs(𝐹)‘𝑥))
22	20, 21	eqtrid 2809	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (cf‘𝐴) → (𝐺‘𝑥) = (recs(𝐹)‘𝑥))
23		recsval 8375	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ On → (recs(𝐹)‘𝑥) = (𝐹‘(recs(𝐹) ↾ 𝑥)))
24		recsfnon 8374	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ recs(𝐹) Fn On
25		fnfun 6621	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (recs(𝐹) Fn On → Fun recs(𝐹))
26	24, 25	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Fun recs(𝐹)
27		vex 3458	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑥 ∈ V
28		resfunexg 7199	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Fun recs(𝐹) ∧ 𝑥 ∈ V) → (recs(𝐹) ↾ 𝑥) ∈ V)
29	26, 27, 28	mp2an 702	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (recs(𝐹) ↾ 𝑥) ∈ V
30		dmeq 5879	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 = (recs(𝐹) ↾ 𝑥) → dom 𝑧 = dom (recs(𝐹) ↾ 𝑥))
31	30	fveq2d 6871	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 = (recs(𝐹) ↾ 𝑥) → (𝑔‘dom 𝑧) = (𝑔‘dom (recs(𝐹) ↾ 𝑥)))
32		fveq1 6866	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 = (recs(𝐹) ↾ 𝑥) → (𝑧‘𝑡) = ((recs(𝐹) ↾ 𝑥)‘𝑡))
33		suceq 6414	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑧‘𝑡) = ((recs(𝐹) ↾ 𝑥)‘𝑡) → suc (𝑧‘𝑡) = suc ((recs(𝐹) ↾ 𝑥)‘𝑡))
34	32, 33	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 = (recs(𝐹) ↾ 𝑥) → suc (𝑧‘𝑡) = suc ((recs(𝐹) ↾ 𝑥)‘𝑡))
35	30, 34	iuneq12d 4979	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 = (recs(𝐹) ↾ 𝑥) → ∪ 𝑡 ∈ dom 𝑧 suc (𝑧‘𝑡) = ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡))
36	31, 35	uneq12d 4122	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = (recs(𝐹) ↾ 𝑥) → ((𝑔‘dom 𝑧) ∪ ∪ 𝑡 ∈ dom 𝑧 suc (𝑧‘𝑡)) = ((𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∪ ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡)))
37		cfsmolem.2	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐹 = (𝑧 ∈ V ↦ ((𝑔‘dom 𝑧) ∪ ∪ 𝑡 ∈ dom 𝑧 suc (𝑧‘𝑡)))
38		fvex 6880	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∈ V
39	29	dmex 7890	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ dom (recs(𝐹) ↾ 𝑥) ∈ V
40		fvex 6880	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((recs(𝐹) ↾ 𝑥)‘𝑡) ∈ V
41	40	sucex 7789	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ suc ((recs(𝐹) ↾ 𝑥)‘𝑡) ∈ V
42	39, 41	iunex 7949	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡) ∈ V
43	38, 42	unex 7727	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∪ ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡)) ∈ V
44	36, 37, 43	fvmpt 6975	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((recs(𝐹) ↾ 𝑥) ∈ V → (𝐹‘(recs(𝐹) ↾ 𝑥)) = ((𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∪ ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡)))
45	29, 44	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹‘(recs(𝐹) ↾ 𝑥)) = ((𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∪ ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡))
46	23, 45	eqtrdi 2813	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ On → (recs(𝐹)‘𝑥) = ((𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∪ ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡)))
47		onss 7768	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
48		fnssres 6644	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((recs(𝐹) Fn On ∧ 𝑥 ⊆ On) → (recs(𝐹) ↾ 𝑥) Fn 𝑥)
49	24, 47, 48	sylancr 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ On → (recs(𝐹) ↾ 𝑥) Fn 𝑥)
50		fndm 6624	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((recs(𝐹) ↾ 𝑥) Fn 𝑥 → dom (recs(𝐹) ↾ 𝑥) = 𝑥)
51		fveq2 6867	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (dom (recs(𝐹) ↾ 𝑥) = 𝑥 → (𝑔‘dom (recs(𝐹) ↾ 𝑥)) = (𝑔‘𝑥))
52		iuneq1 4966	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (dom (recs(𝐹) ↾ 𝑥) = 𝑥 → ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡) = ∪ 𝑡 ∈ 𝑥 suc ((recs(𝐹) ↾ 𝑥)‘𝑡))
53		fvres 6886	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 ∈ 𝑥 → ((recs(𝐹) ↾ 𝑥)‘𝑡) = (recs(𝐹)‘𝑡))
54		suceq 6414	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((recs(𝐹) ↾ 𝑥)‘𝑡) = (recs(𝐹)‘𝑡) → suc ((recs(𝐹) ↾ 𝑥)‘𝑡) = suc (recs(𝐹)‘𝑡))
55	53, 54	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 ∈ 𝑥 → suc ((recs(𝐹) ↾ 𝑥)‘𝑡) = suc (recs(𝐹)‘𝑡))
56	55	iuneq2i 4971	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ∪ 𝑡 ∈ 𝑥 suc ((recs(𝐹) ↾ 𝑥)‘𝑡) = ∪ 𝑡 ∈ 𝑥 suc (recs(𝐹)‘𝑡)
57		fveq2 6867	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = 𝑡 → (recs(𝐹)‘𝑦) = (recs(𝐹)‘𝑡))
58		suceq 6414	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((recs(𝐹)‘𝑦) = (recs(𝐹)‘𝑡) → suc (recs(𝐹)‘𝑦) = suc (recs(𝐹)‘𝑡))
59	57, 58	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑡 → suc (recs(𝐹)‘𝑦) = suc (recs(𝐹)‘𝑡))
60	59	cbviunv 4996	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = ∪ 𝑡 ∈ 𝑥 suc (recs(𝐹)‘𝑡)
61	56, 60	eqtr4i 2788	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∪ 𝑡 ∈ 𝑥 suc ((recs(𝐹) ↾ 𝑥)‘𝑡) = ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦)
62	52, 61	eqtrdi 2813	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (dom (recs(𝐹) ↾ 𝑥) = 𝑥 → ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡) = ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦))
63	51, 62	uneq12d 4122	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (dom (recs(𝐹) ↾ 𝑥) = 𝑥 → ((𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∪ ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡)) = ((𝑔‘𝑥) ∪ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦)))
64	49, 50, 63	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ On → ((𝑔‘dom (recs(𝐹) ↾ 𝑥)) ∪ ∪ 𝑡 ∈ dom (recs(𝐹) ↾ 𝑥)suc ((recs(𝐹) ↾ 𝑥)‘𝑡)) = ((𝑔‘𝑥) ∪ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦)))
65	46, 64	eqtrd 2797	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ On → (recs(𝐹)‘𝑥) = ((𝑔‘𝑥) ∪ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦)))
66	3, 65	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (cf‘𝐴) → (recs(𝐹)‘𝑥) = ((𝑔‘𝑥) ∪ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦)))
67	22, 66	eqtrd 2797	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (cf‘𝐴) → (𝐺‘𝑥) = ((𝑔‘𝑥) ∪ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦)))
68	67	3ad2ant2 1147	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → (𝐺‘𝑥) = ((𝑔‘𝑥) ∪ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦)))
69		eloni 6356	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ On → Ord 𝐴)
70	69	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On) → Ord 𝐴)
71	70	3ad2ant1 1146	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → Ord 𝐴)
72		f1f 6760	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔:(cf‘𝐴)–1-1→𝐴 → 𝑔:(cf‘𝐴)⟶𝐴)
73	72	ffvelcdmda 7065	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝑥 ∈ (cf‘𝐴)) → (𝑔‘𝑥) ∈ 𝐴)
74	73	adantlr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴)) → (𝑔‘𝑥) ∈ 𝐴)
75	74	3adant3 1145	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → (𝑔‘𝑥) ∈ 𝐴)
76	19	fveq1i 6868	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐺‘𝑦) = ((recs(𝐹) ↾ (cf‘𝐴))‘𝑦)
77	13	fvresd 6887	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (cf‘𝐴)) → ((recs(𝐹) ↾ (cf‘𝐴))‘𝑦) = (recs(𝐹)‘𝑦))
78	76, 77	eqtrid 2809	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (cf‘𝐴)) → (𝐺‘𝑦) = (recs(𝐹)‘𝑦))
79	78	adantrl 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ 𝑥 ∧ (𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴))) → (𝐺‘𝑦) = (recs(𝐹)‘𝑦))
80	79	ancoms 462	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦 ∈ 𝑥) → (𝐺‘𝑦) = (recs(𝐹)‘𝑦))
81	80	eleq1d 2847	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦 ∈ 𝑥) → ((𝐺‘𝑦) ∈ 𝐴 ↔ (recs(𝐹)‘𝑦) ∈ 𝐴))
82		ordsucss 7798	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (Ord 𝐴 → ((recs(𝐹)‘𝑦) ∈ 𝐴 → suc (recs(𝐹)‘𝑦) ⊆ 𝐴))
83	69, 82	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ∈ On → ((recs(𝐹)‘𝑦) ∈ 𝐴 → suc (recs(𝐹)‘𝑦) ⊆ 𝐴))
84	83	ad2antrr 736	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦 ∈ 𝑥) → ((recs(𝐹)‘𝑦) ∈ 𝐴 → suc (recs(𝐹)‘𝑦) ⊆ 𝐴))
85	81, 84	sylbid 242	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦 ∈ 𝑥) → ((𝐺‘𝑦) ∈ 𝐴 → suc (recs(𝐹)‘𝑦) ⊆ 𝐴))
86	85	ralimdva 3174	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴 → ∀𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴))
87		iunss 5002	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴 ↔ ∀𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴)
88	86, 87	imbitrrdi 254	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴 → ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴))
89	88	3impia 1130	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴)
90		onelon 6371	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐴 ∈ On ∧ (recs(𝐹)‘𝑦) ∈ 𝐴) → (recs(𝐹)‘𝑦) ∈ On)
91	90	ex 416	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐴 ∈ On → ((recs(𝐹)‘𝑦) ∈ 𝐴 → (recs(𝐹)‘𝑦) ∈ On))
92	91	ad2antrr 736	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦 ∈ 𝑥) → ((recs(𝐹)‘𝑦) ∈ 𝐴 → (recs(𝐹)‘𝑦) ∈ On))
93	81, 92	sylbid 242	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦 ∈ 𝑥) → ((𝐺‘𝑦) ∈ 𝐴 → (recs(𝐹)‘𝑦) ∈ On))
94		onsuc 7793	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((recs(𝐹)‘𝑦) ∈ On → suc (recs(𝐹)‘𝑦) ∈ On)
95	93, 94	syl6 35	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) ∧ 𝑦 ∈ 𝑥) → ((𝐺‘𝑦) ∈ 𝐴 → suc (recs(𝐹)‘𝑦) ∈ On))
96	95	ralimdva 3174	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴 → ∀𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ∈ On))
97	96	3impia 1130	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → ∀𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ∈ On)
98		iunon 8310	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ∈ On) → ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ∈ On)
99	27, 97, 98	sylancr 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ∈ On)
100		simp1 1149	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → 𝐴 ∈ On)
101		onsseleq 6387	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ∈ On ∧ 𝐴 ∈ On) → (∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴 ↔ (∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴 ∨ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴)))
102	99, 100, 101	syl2anc 593	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → (∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴 ↔ (∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴 ∨ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴)))
103		idd 24	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → (∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴 → ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴))
104		simpll 776	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) ∧ (∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ 𝐴 ∈ On)) → 𝑥 ∈ (cf‘𝐴))
105		simprr 782	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) ∧ (∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ 𝐴 ∈ On)) → 𝐴 ∈ On)
106	3	ad2antrr 736	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) ∧ (∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ 𝐴 ∈ On)) → 𝑥 ∈ On)
107	3, 49	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 ∈ (cf‘𝐴) → (recs(𝐹) ↾ 𝑥) Fn 𝑥)
108	107	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → (recs(𝐹) ↾ 𝑥) Fn 𝑥)
109	78	ancoms 462	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑥 ∈ (cf‘𝐴) ∧ 𝑦 ∈ 𝑥) → (𝐺‘𝑦) = (recs(𝐹)‘𝑦))
110		fvres 6886	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑦 ∈ 𝑥 → ((recs(𝐹) ↾ 𝑥)‘𝑦) = (recs(𝐹)‘𝑦))
111	110	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑥 ∈ (cf‘𝐴) ∧ 𝑦 ∈ 𝑥) → ((recs(𝐹) ↾ 𝑥)‘𝑦) = (recs(𝐹)‘𝑦))
112	109, 111	eqtr4d 2800	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑥 ∈ (cf‘𝐴) ∧ 𝑦 ∈ 𝑥) → (𝐺‘𝑦) = ((recs(𝐹) ↾ 𝑥)‘𝑦))
113	112	eleq1d 2847	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑥 ∈ (cf‘𝐴) ∧ 𝑦 ∈ 𝑥) → ((𝐺‘𝑦) ∈ 𝐴 ↔ ((recs(𝐹) ↾ 𝑥)‘𝑦) ∈ 𝐴))
114	113	ralbidva 3183	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 ∈ (cf‘𝐴) → (∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴 ↔ ∀𝑦 ∈ 𝑥 ((recs(𝐹) ↾ 𝑥)‘𝑦) ∈ 𝐴))
115	114	biimpa 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → ∀𝑦 ∈ 𝑥 ((recs(𝐹) ↾ 𝑥)‘𝑦) ∈ 𝐴)
116		ffnfv 7100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((recs(𝐹) ↾ 𝑥):𝑥⟶𝐴 ↔ ((recs(𝐹) ↾ 𝑥) Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 ((recs(𝐹) ↾ 𝑥)‘𝑦) ∈ 𝐴))
117	108, 115, 116	sylanbrc 592	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → (recs(𝐹) ↾ 𝑥):𝑥⟶𝐴)
118		eleq2 2851	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 → (𝑡 ∈ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ↔ 𝑡 ∈ 𝐴))
119	118	biimpar 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ 𝑡 ∈ 𝐴) → 𝑡 ∈ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦))
120	119	adantrl 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ (𝐴 ∈ On ∧ 𝑡 ∈ 𝐴)) → 𝑡 ∈ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦))
121	120	3adant1 1143	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((recs(𝐹) ↾ 𝑥):𝑥⟶𝐴 ∧ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ (𝐴 ∈ On ∧ 𝑡 ∈ 𝐴)) → 𝑡 ∈ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦))
122		onelon 6371	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((𝐴 ∈ On ∧ 𝑡 ∈ 𝐴) → 𝑡 ∈ On)
123	110	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (((recs(𝐹) ↾ 𝑥):𝑥⟶𝐴 ∧ 𝑦 ∈ 𝑥) → ((recs(𝐹) ↾ 𝑥)‘𝑦) = (recs(𝐹)‘𝑦))
124		ffvelcdm 7062	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (((recs(𝐹) ↾ 𝑥):𝑥⟶𝐴 ∧ 𝑦 ∈ 𝑥) → ((recs(𝐹) ↾ 𝑥)‘𝑦) ∈ 𝐴)
125	123, 124	eqeltrrd 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (((recs(𝐹) ↾ 𝑥):𝑥⟶𝐴 ∧ 𝑦 ∈ 𝑥) → (recs(𝐹)‘𝑦) ∈ 𝐴)
126	125, 90	sylan2 602	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((𝐴 ∈ On ∧ ((recs(𝐹) ↾ 𝑥):𝑥⟶𝐴 ∧ 𝑦 ∈ 𝑥)) → (recs(𝐹)‘𝑦) ∈ On)
127	126	adantlr 725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((𝐴 ∈ On ∧ 𝑡 ∈ 𝐴) ∧ ((recs(𝐹) ↾ 𝑥):𝑥⟶𝐴 ∧ 𝑦 ∈ 𝑥)) → (recs(𝐹)‘𝑦) ∈ On)
128		onsssuc 6438	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((𝑡 ∈ On ∧ (recs(𝐹)‘𝑦) ∈ On) → (𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 ∈ suc (recs(𝐹)‘𝑦)))
129	122, 127, 128	syl2an2r 695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((𝐴 ∈ On ∧ 𝑡 ∈ 𝐴) ∧ ((recs(𝐹) ↾ 𝑥):𝑥⟶𝐴 ∧ 𝑦 ∈ 𝑥)) → (𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 ∈ suc (recs(𝐹)‘𝑦)))
130	129	anassrs 471	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((𝐴 ∈ On ∧ 𝑡 ∈ 𝐴) ∧ (recs(𝐹) ↾ 𝑥):𝑥⟶𝐴) ∧ 𝑦 ∈ 𝑥) → (𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 ∈ suc (recs(𝐹)‘𝑦)))
131	130	rexbidva 3184	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝐴 ∈ On ∧ 𝑡 ∈ 𝐴) ∧ (recs(𝐹) ↾ 𝑥):𝑥⟶𝐴) → (∃𝑦 ∈ 𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ ∃𝑦 ∈ 𝑥 𝑡 ∈ suc (recs(𝐹)‘𝑦)))
132		eliun 4953	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑡 ∈ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ↔ ∃𝑦 ∈ 𝑥 𝑡 ∈ suc (recs(𝐹)‘𝑦))
133	131, 132	bitr4di 291	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝐴 ∈ On ∧ 𝑡 ∈ 𝐴) ∧ (recs(𝐹) ↾ 𝑥):𝑥⟶𝐴) → (∃𝑦 ∈ 𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 ∈ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦)))
134	133	ancoms 462	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((recs(𝐹) ↾ 𝑥):𝑥⟶𝐴 ∧ (𝐴 ∈ On ∧ 𝑡 ∈ 𝐴)) → (∃𝑦 ∈ 𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 ∈ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦)))
135	134	3adant2 1144	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((recs(𝐹) ↾ 𝑥):𝑥⟶𝐴 ∧ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ (𝐴 ∈ On ∧ 𝑡 ∈ 𝐴)) → (∃𝑦 ∈ 𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦) ↔ 𝑡 ∈ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦)))
136	121, 135	mpbird 259	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((recs(𝐹) ↾ 𝑥):𝑥⟶𝐴 ∧ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ (𝐴 ∈ On ∧ 𝑡 ∈ 𝐴)) → ∃𝑦 ∈ 𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
137	136	3expa 1131	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((recs(𝐹) ↾ 𝑥):𝑥⟶𝐴 ∧ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴) ∧ (𝐴 ∈ On ∧ 𝑡 ∈ 𝐴)) → ∃𝑦 ∈ 𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
138	137	anassrs 471	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((recs(𝐹) ↾ 𝑥):𝑥⟶𝐴 ∧ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴) ∧ 𝐴 ∈ On) ∧ 𝑡 ∈ 𝐴) → ∃𝑦 ∈ 𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
139	138	ralrimiva 3154	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((recs(𝐹) ↾ 𝑥):𝑥⟶𝐴 ∧ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴) ∧ 𝐴 ∈ On) → ∀𝑡 ∈ 𝐴 ∃𝑦 ∈ 𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
140	139	expl 461	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((recs(𝐹) ↾ 𝑥):𝑥⟶𝐴 → ((∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ 𝐴 ∈ On) → ∀𝑡 ∈ 𝐴 ∃𝑦 ∈ 𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦)))
141	117, 140	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → ((∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ 𝐴 ∈ On) → ∀𝑡 ∈ 𝐴 ∃𝑦 ∈ 𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦)))
142	141	imp 410	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) ∧ (∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ 𝐴 ∈ On)) → ∀𝑡 ∈ 𝐴 ∃𝑦 ∈ 𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
143		feq1 6669	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓 = (recs(𝐹) ↾ 𝑥) → (𝑓:𝑥⟶𝐴 ↔ (recs(𝐹) ↾ 𝑥):𝑥⟶𝐴))
144		fveq1 6866	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑓 = (recs(𝐹) ↾ 𝑥) → (𝑓‘𝑦) = ((recs(𝐹) ↾ 𝑥)‘𝑦))
145	144	sseq2d 3968	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑓 = (recs(𝐹) ↾ 𝑥) → (𝑡 ⊆ (𝑓‘𝑦) ↔ 𝑡 ⊆ ((recs(𝐹) ↾ 𝑥)‘𝑦)))
146	145	rexbidv 3186	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑓 = (recs(𝐹) ↾ 𝑥) → (∃𝑦 ∈ 𝑥 𝑡 ⊆ (𝑓‘𝑦) ↔ ∃𝑦 ∈ 𝑥 𝑡 ⊆ ((recs(𝐹) ↾ 𝑥)‘𝑦)))
147	110	sseq2d 3968	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑦 ∈ 𝑥 → (𝑡 ⊆ ((recs(𝐹) ↾ 𝑥)‘𝑦) ↔ 𝑡 ⊆ (recs(𝐹)‘𝑦)))
148	147	rexbiia 3107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (∃𝑦 ∈ 𝑥 𝑡 ⊆ ((recs(𝐹) ↾ 𝑥)‘𝑦) ↔ ∃𝑦 ∈ 𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))
149	146, 148	bitrdi 289	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓 = (recs(𝐹) ↾ 𝑥) → (∃𝑦 ∈ 𝑥 𝑡 ⊆ (𝑓‘𝑦) ↔ ∃𝑦 ∈ 𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦)))
150	149	ralbidv 3185	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓 = (recs(𝐹) ↾ 𝑥) → (∀𝑡 ∈ 𝐴 ∃𝑦 ∈ 𝑥 𝑡 ⊆ (𝑓‘𝑦) ↔ ∀𝑡 ∈ 𝐴 ∃𝑦 ∈ 𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦)))
151	143, 150	anbi12d 641	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓 = (recs(𝐹) ↾ 𝑥) → ((𝑓:𝑥⟶𝐴 ∧ ∀𝑡 ∈ 𝐴 ∃𝑦 ∈ 𝑥 𝑡 ⊆ (𝑓‘𝑦)) ↔ ((recs(𝐹) ↾ 𝑥):𝑥⟶𝐴 ∧ ∀𝑡 ∈ 𝐴 ∃𝑦 ∈ 𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦))))
152	29, 151	spcev 3565	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((recs(𝐹) ↾ 𝑥):𝑥⟶𝐴 ∧ ∀𝑡 ∈ 𝐴 ∃𝑦 ∈ 𝑥 𝑡 ⊆ (recs(𝐹)‘𝑦)) → ∃𝑓(𝑓:𝑥⟶𝐴 ∧ ∀𝑡 ∈ 𝐴 ∃𝑦 ∈ 𝑥 𝑡 ⊆ (𝑓‘𝑦)))
153	117, 142, 152	syl2an2r 695	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) ∧ (∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ 𝐴 ∈ On)) → ∃𝑓(𝑓:𝑥⟶𝐴 ∧ ∀𝑡 ∈ 𝐴 ∃𝑦 ∈ 𝑥 𝑡 ⊆ (𝑓‘𝑦)))
154		cfflb 10216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∃𝑓(𝑓:𝑥⟶𝐴 ∧ ∀𝑡 ∈ 𝐴 ∃𝑦 ∈ 𝑥 𝑡 ⊆ (𝑓‘𝑦)) → (cf‘𝐴) ⊆ 𝑥))
155	154	imp 410	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∃𝑓(𝑓:𝑥⟶𝐴 ∧ ∀𝑡 ∈ 𝐴 ∃𝑦 ∈ 𝑥 𝑡 ⊆ (𝑓‘𝑦))) → (cf‘𝐴) ⊆ 𝑥)
156	105, 106, 153, 155	syl21anc 848	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) ∧ (∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ 𝐴 ∈ On)) → (cf‘𝐴) ⊆ 𝑥)
157		ontri1 6380	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((cf‘𝐴) ∈ On ∧ 𝑥 ∈ On) → ((cf‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (cf‘𝐴)))
158	2, 3, 157	sylancr 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 ∈ (cf‘𝐴) → ((cf‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (cf‘𝐴)))
159	158	ad2antrr 736	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) ∧ (∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ 𝐴 ∈ On)) → ((cf‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (cf‘𝐴)))
160	156, 159	mpbid 234	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) ∧ (∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ 𝐴 ∈ On)) → ¬ 𝑥 ∈ (cf‘𝐴))
161	104, 160	pm2.21dd 197	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) ∧ (∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ 𝐴 ∈ On)) → ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴)
162	161	ex 416	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → ((∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 ∧ 𝐴 ∈ On) → ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴))
163	162	expcomd 420	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → (𝐴 ∈ On → (∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 → ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴)))
164	163	com12 32	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ∈ On → ((𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → (∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 → ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴)))
165	164	3impib 1129	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → (∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴 → ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴))
166	103, 165	jaod 870	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → ((∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴 ∨ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) = 𝐴) → ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴))
167	102, 166	sylbid 242	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → (∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ⊆ 𝐴 → ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴))
168	89, 167	mpd 15	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴)
169	168	3adant1l 1190	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴)
170		ordunel 7807	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝐴 ∧ (𝑔‘𝑥) ∈ 𝐴 ∧ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) ∈ 𝐴) → ((𝑔‘𝑥) ∪ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦)) ∈ 𝐴)
171	71, 75, 169, 170	syl3anc 1390	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → ((𝑔‘𝑥) ∪ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦)) ∈ 𝐴)
172	68, 171	eqeltrd 2862	. . . . . . . . . . . . . . 15 ⊢ (((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴) ∧ ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐴)
173	172	3expia 1134	. . . . . . . . . . . . . 14 ⊢ (((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On) ∧ 𝑥 ∈ (cf‘𝐴)) → (∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴 → (𝐺‘𝑥) ∈ 𝐴))
174	173	3impa 1122	. . . . . . . . . . . . 13 ⊢ ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ 𝐴 → (𝐺‘𝑥) ∈ 𝐴))
175	18, 174	syldc 48	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝑥 ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On ∧ 𝑦 ∈ (cf‘𝐴)) → (𝐺‘𝑦) ∈ 𝐴) → ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (𝐺‘𝑥) ∈ 𝐴))
176	175	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On ∧ 𝑦 ∈ (cf‘𝐴)) → (𝐺‘𝑦) ∈ 𝐴) → ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (𝐺‘𝑥) ∈ 𝐴)))
177	9, 176	tfis2 7837	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (𝐺‘𝑥) ∈ 𝐴))
178	4, 177	mpcom 38	. . . . . . . . 9 ⊢ ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ (cf‘𝐴)) → (𝐺‘𝑥) ∈ 𝐴)
179	178	3expia 1134	. . . . . . . 8 ⊢ ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On) → (𝑥 ∈ (cf‘𝐴) → (𝐺‘𝑥) ∈ 𝐴))
180	179	ralrimiv 3153	. . . . . . 7 ⊢ ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On) → ∀𝑥 ∈ (cf‘𝐴)(𝐺‘𝑥) ∈ 𝐴)
181	2	onssi 7818	. . . . . . . . 9 ⊢ (cf‘𝐴) ⊆ On
182		fnssres 6644	. . . . . . . . . 10 ⊢ ((recs(𝐹) Fn On ∧ (cf‘𝐴) ⊆ On) → (recs(𝐹) ↾ (cf‘𝐴)) Fn (cf‘𝐴))
183	19	fneq1i 6618	. . . . . . . . . 10 ⊢ (𝐺 Fn (cf‘𝐴) ↔ (recs(𝐹) ↾ (cf‘𝐴)) Fn (cf‘𝐴))
184	182, 183	sylibr 236	. . . . . . . . 9 ⊢ ((recs(𝐹) Fn On ∧ (cf‘𝐴) ⊆ On) → 𝐺 Fn (cf‘𝐴))
185	24, 181, 184	mp2an 702	. . . . . . . 8 ⊢ 𝐺 Fn (cf‘𝐴)
186		ffnfv 7100	. . . . . . . 8 ⊢ (𝐺:(cf‘𝐴)⟶𝐴 ↔ (𝐺 Fn (cf‘𝐴) ∧ ∀𝑥 ∈ (cf‘𝐴)(𝐺‘𝑥) ∈ 𝐴))
187	185, 186	mpbiran 719	. . . . . . 7 ⊢ (𝐺:(cf‘𝐴)⟶𝐴 ↔ ∀𝑥 ∈ (cf‘𝐴)(𝐺‘𝑥) ∈ 𝐴)
188	180, 187	sylibr 236	. . . . . 6 ⊢ ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On) → 𝐺:(cf‘𝐴)⟶𝐴)
189	188	adantlr 725	. . . . 5 ⊢ (((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔‘𝑤)) ∧ 𝐴 ∈ On) → 𝐺:(cf‘𝐴)⟶𝐴)
190		onss 7768	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
191	190	adantl 485	. . . . . . 7 ⊢ ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On) → 𝐴 ⊆ On)
192	2	onordi 6459	. . . . . . . 8 ⊢ Ord (cf‘𝐴)
193		fvex 6880	. . . . . . . . . . . . . . . . 17 ⊢ (recs(𝐹)‘𝑦) ∈ V
194	193	sucid 6430	. . . . . . . . . . . . . . . 16 ⊢ (recs(𝐹)‘𝑦) ∈ suc (recs(𝐹)‘𝑦)
195		fveq2 6867	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑦 → (recs(𝐹)‘𝑡) = (recs(𝐹)‘𝑦))
196		suceq 6414	. . . . . . . . . . . . . . . . . . 19 ⊢ ((recs(𝐹)‘𝑡) = (recs(𝐹)‘𝑦) → suc (recs(𝐹)‘𝑡) = suc (recs(𝐹)‘𝑦))
197	195, 196	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑦 → suc (recs(𝐹)‘𝑡) = suc (recs(𝐹)‘𝑦))
198	197	eliuni 4955	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝑥 ∧ (recs(𝐹)‘𝑦) ∈ suc (recs(𝐹)‘𝑦)) → (recs(𝐹)‘𝑦) ∈ ∪ 𝑡 ∈ 𝑥 suc (recs(𝐹)‘𝑡))
199	198, 60	eleqtrrdi 2873	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝑥 ∧ (recs(𝐹)‘𝑦) ∈ suc (recs(𝐹)‘𝑦)) → (recs(𝐹)‘𝑦) ∈ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦))
200	194, 199	mpan2 701	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝑥 → (recs(𝐹)‘𝑦) ∈ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦))
201		elun2 4135	. . . . . . . . . . . . . . 15 ⊢ ((recs(𝐹)‘𝑦) ∈ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦) → (recs(𝐹)‘𝑦) ∈ ((𝑔‘𝑥) ∪ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦)))
202	200, 201	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝑥 → (recs(𝐹)‘𝑦) ∈ ((𝑔‘𝑥) ∪ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦)))
203	202	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (cf‘𝐴)) → (recs(𝐹)‘𝑦) ∈ ((𝑔‘𝑥) ∪ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦)))
204	3	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (cf‘𝐴)) → 𝑥 ∈ On)
205	204, 65	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (cf‘𝐴)) → (recs(𝐹)‘𝑥) = ((𝑔‘𝑥) ∪ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦)))
206	203, 205	eleqtrrd 2865	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (cf‘𝐴)) → (recs(𝐹)‘𝑦) ∈ (recs(𝐹)‘𝑥))
207	22	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (cf‘𝐴)) → (𝐺‘𝑥) = (recs(𝐹)‘𝑥))
208	206, 78, 207	3eltr4d 2877	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (cf‘𝐴)) → (𝐺‘𝑦) ∈ (𝐺‘𝑥))
209	208	expcom 417	. . . . . . . . . 10 ⊢ (𝑥 ∈ (cf‘𝐴) → (𝑦 ∈ 𝑥 → (𝐺‘𝑦) ∈ (𝐺‘𝑥)))
210	209	ralrimiv 3153	. . . . . . . . 9 ⊢ (𝑥 ∈ (cf‘𝐴) → ∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ (𝐺‘𝑥))
211	210	rgen 3078	. . . . . . . 8 ⊢ ∀𝑥 ∈ (cf‘𝐴)∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ (𝐺‘𝑥)
212		issmo2 8320	. . . . . . . . 9 ⊢ (𝐺:(cf‘𝐴)⟶𝐴 → ((𝐴 ⊆ On ∧ Ord (cf‘𝐴) ∧ ∀𝑥 ∈ (cf‘𝐴)∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ (𝐺‘𝑥)) → Smo 𝐺))
213	212	com12 32	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ Ord (cf‘𝐴) ∧ ∀𝑥 ∈ (cf‘𝐴)∀𝑦 ∈ 𝑥 (𝐺‘𝑦) ∈ (𝐺‘𝑥)) → (𝐺:(cf‘𝐴)⟶𝐴 → Smo 𝐺))
214	192, 211, 213	mp3an23 1474	. . . . . . 7 ⊢ (𝐴 ⊆ On → (𝐺:(cf‘𝐴)⟶𝐴 → Smo 𝐺))
215	191, 188, 214	sylc 65	. . . . . 6 ⊢ ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ 𝐴 ∈ On) → Smo 𝐺)
216	215	adantlr 725	. . . . 5 ⊢ (((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔‘𝑤)) ∧ 𝐴 ∈ On) → Smo 𝐺)
217		fveq2 6867	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝑔‘𝑥) = (𝑔‘𝑤))
218		fveq2 6867	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝐺‘𝑥) = (𝐺‘𝑤))
219	217, 218	sseq12d 3969	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → ((𝑔‘𝑥) ⊆ (𝐺‘𝑥) ↔ (𝑔‘𝑤) ⊆ (𝐺‘𝑤)))
220		ssun1 4130	. . . . . . . . . . 11 ⊢ (𝑔‘𝑥) ⊆ ((𝑔‘𝑥) ∪ ∪ 𝑦 ∈ 𝑥 suc (recs(𝐹)‘𝑦))
221	220, 67	sseqtrrid 3979	. . . . . . . . . 10 ⊢ (𝑥 ∈ (cf‘𝐴) → (𝑔‘𝑥) ⊆ (𝐺‘𝑥))
222	219, 221	vtoclga 3541	. . . . . . . . 9 ⊢ (𝑤 ∈ (cf‘𝐴) → (𝑔‘𝑤) ⊆ (𝐺‘𝑤))
223		sstr 3944	. . . . . . . . . 10 ⊢ ((𝑧 ⊆ (𝑔‘𝑤) ∧ (𝑔‘𝑤) ⊆ (𝐺‘𝑤)) → 𝑧 ⊆ (𝐺‘𝑤))
224	223	expcom 417	. . . . . . . . 9 ⊢ ((𝑔‘𝑤) ⊆ (𝐺‘𝑤) → (𝑧 ⊆ (𝑔‘𝑤) → 𝑧 ⊆ (𝐺‘𝑤)))
225	222, 224	syl 17	. . . . . . . 8 ⊢ (𝑤 ∈ (cf‘𝐴) → (𝑧 ⊆ (𝑔‘𝑤) → 𝑧 ⊆ (𝐺‘𝑤)))
226	225	reximia 3097	. . . . . . 7 ⊢ (∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔‘𝑤) → ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺‘𝑤))
227	226	ralimi 3099	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔‘𝑤) → ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺‘𝑤))
228	227	ad2antlr 737	. . . . 5 ⊢ (((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔‘𝑤)) ∧ 𝐴 ∈ On) → ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺‘𝑤))
229		fnex 7201	. . . . . . 7 ⊢ ((𝐺 Fn (cf‘𝐴) ∧ (cf‘𝐴) ∈ On) → 𝐺 ∈ V)
230	185, 2, 229	mp2an 702	. . . . . 6 ⊢ 𝐺 ∈ V
231		feq1 6669	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (𝑓:(cf‘𝐴)⟶𝐴 ↔ 𝐺:(cf‘𝐴)⟶𝐴))
232		smoeq 8321	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (Smo 𝑓 ↔ Smo 𝐺))
233		fveq1 6866	. . . . . . . . . 10 ⊢ (𝑓 = 𝐺 → (𝑓‘𝑤) = (𝐺‘𝑤))
234	233	sseq2d 3968	. . . . . . . . 9 ⊢ (𝑓 = 𝐺 → (𝑧 ⊆ (𝑓‘𝑤) ↔ 𝑧 ⊆ (𝐺‘𝑤)))
235	234	rexbidv 3186	. . . . . . . 8 ⊢ (𝑓 = 𝐺 → (∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤) ↔ ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺‘𝑤)))
236	235	ralbidv 3185	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤) ↔ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺‘𝑤)))
237	231, 232, 236	3anbi123d 1457	. . . . . 6 ⊢ (𝑓 = 𝐺 → ((𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)) ↔ (𝐺:(cf‘𝐴)⟶𝐴 ∧ Smo 𝐺 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺‘𝑤))))
238	230, 237	spcev 3565	. . . . 5 ⊢ ((𝐺:(cf‘𝐴)⟶𝐴 ∧ Smo 𝐺 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝐺‘𝑤)) → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))
239	189, 216, 228, 238	syl3anc 1390	. . . 4 ⊢ (((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔‘𝑤)) ∧ 𝐴 ∈ On) → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))
240	239	expcom 417	. . 3 ⊢ (𝐴 ∈ On → ((𝑔:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔‘𝑤)) → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤))))
241	240	exlimdv 1953	. 2 ⊢ (𝐴 ∈ On → (∃𝑔(𝑔:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑔‘𝑤)) → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤))))
242	1, 241	mpd 15	1 ⊢ (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1098   = wceq 1560  ∃wex 1799   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086  Vcvv 3454   ∪ cun 3902   ⊆ wss 3904  ∪ ciun 4949   ↦ cmpt 5181  dom cdm 5647   ↾ cres 5649  Ord word 6345  Oncon0 6346  suc csuc 6348  Fun wfun 6515   Fn wfn 6516  ⟶wf 6517  –1-1→wf1 6518  ‘cfv 6521  Smo wsmo 8316  recscrecs 8341  cfccf 9895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-smo 8317  df-recs 8342  df-er 8678  df-map 8810  df-en 8928  df-dom 8929  df-sdom 8930  df-card 9897  df-cf 9899  df-acn 9900

This theorem is referenced by:  cfsmo  10228