Theorem sornom 10206

Description: The range of a single-step monotone function from ω into a partially ordered set is a chain. (Contributed by Stefan O'Rear, 3-Nov-2014.)

Assertion

Ref	Expression
sornom	⊢ ((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → 𝑅 Or ran 𝐹)

Distinct variable groups:   𝐹,𝑎   𝑅,𝑎

Proof of Theorem sornom

Dummy variables 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1138	. 2 ⊢ ((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → 𝑅 Po ran 𝐹)
2		fvelrnb 6903	. . . . . 6 ⊢ (𝐹 Fn ω → (𝑏 ∈ ran 𝐹 ↔ ∃𝑑 ∈ ω (𝐹‘𝑑) = 𝑏))
3		fvelrnb 6903	. . . . . 6 ⊢ (𝐹 Fn ω → (𝑐 ∈ ran 𝐹 ↔ ∃𝑒 ∈ ω (𝐹‘𝑒) = 𝑐))
4	2, 3	anbi12d 632	. . . . 5 ⊢ (𝐹 Fn ω → ((𝑏 ∈ ran 𝐹 ∧ 𝑐 ∈ ran 𝐹) ↔ (∃𝑑 ∈ ω (𝐹‘𝑑) = 𝑏 ∧ ∃𝑒 ∈ ω (𝐹‘𝑒) = 𝑐)))
5	4	3ad2ant1 1133	. . . 4 ⊢ ((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → ((𝑏 ∈ ran 𝐹 ∧ 𝑐 ∈ ran 𝐹) ↔ (∃𝑑 ∈ ω (𝐹‘𝑑) = 𝑏 ∧ ∃𝑒 ∈ ω (𝐹‘𝑒) = 𝑐)))
6		reeanv 3207	. . . . 5 ⊢ (∃𝑑 ∈ ω ∃𝑒 ∈ ω ((𝐹‘𝑑) = 𝑏 ∧ (𝐹‘𝑒) = 𝑐) ↔ (∃𝑑 ∈ ω (𝐹‘𝑑) = 𝑏 ∧ ∃𝑒 ∈ ω (𝐹‘𝑒) = 𝑐))
7		nnord 7830	. . . . . . . . . . 11 ⊢ (𝑑 ∈ ω → Ord 𝑑)
8		nnord 7830	. . . . . . . . . . 11 ⊢ (𝑒 ∈ ω → Ord 𝑒)
9		ordtri2or2 6421	. . . . . . . . . . 11 ⊢ ((Ord 𝑑 ∧ Ord 𝑒) → (𝑑 ⊆ 𝑒 ∨ 𝑒 ⊆ 𝑑))
10	7, 8, 9	syl2an 596	. . . . . . . . . 10 ⊢ ((𝑑 ∈ ω ∧ 𝑒 ∈ ω) → (𝑑 ⊆ 𝑒 ∨ 𝑒 ⊆ 𝑑))
11	10	adantl 481	. . . . . . . . 9 ⊢ (((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) ∧ (𝑑 ∈ ω ∧ 𝑒 ∈ ω)) → (𝑑 ⊆ 𝑒 ∨ 𝑒 ⊆ 𝑑))
12		vex 3448	. . . . . . . . . . 11 ⊢ 𝑑 ∈ V
13		vex 3448	. . . . . . . . . . 11 ⊢ 𝑒 ∈ V
14		eleq1w 2811	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑑 → (𝑏 ∈ ω ↔ 𝑑 ∈ ω))
15		eleq1w 2811	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑒 → (𝑐 ∈ ω ↔ 𝑒 ∈ ω))
16	14, 15	bi2anan9 638	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 𝑑 ∧ 𝑐 = 𝑒) → ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) ↔ (𝑑 ∈ ω ∧ 𝑒 ∈ ω)))
17	16	anbi2d 630	. . . . . . . . . . . 12 ⊢ ((𝑏 = 𝑑 ∧ 𝑐 = 𝑒) → (((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ↔ ((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) ∧ (𝑑 ∈ ω ∧ 𝑒 ∈ ω))))
18		sseq12 3971	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 𝑑 ∧ 𝑐 = 𝑒) → (𝑏 ⊆ 𝑐 ↔ 𝑑 ⊆ 𝑒))
19		fveq2 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑑 → (𝐹‘𝑏) = (𝐹‘𝑑))
20		fveq2 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑒 → (𝐹‘𝑐) = (𝐹‘𝑒))
21	19, 20	breqan12d 5118	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 𝑑 ∧ 𝑐 = 𝑒) → ((𝐹‘𝑏)𝑅(𝐹‘𝑐) ↔ (𝐹‘𝑑)𝑅(𝐹‘𝑒)))
22	19, 20	eqeqan12d 2743	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 𝑑 ∧ 𝑐 = 𝑒) → ((𝐹‘𝑏) = (𝐹‘𝑐) ↔ (𝐹‘𝑑) = (𝐹‘𝑒)))
23	21, 22	orbi12d 918	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 𝑑 ∧ 𝑐 = 𝑒) → (((𝐹‘𝑏)𝑅(𝐹‘𝑐) ∨ (𝐹‘𝑏) = (𝐹‘𝑐)) ↔ ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒))))
24	18, 23	imbi12d 344	. . . . . . . . . . . 12 ⊢ ((𝑏 = 𝑑 ∧ 𝑐 = 𝑒) → ((𝑏 ⊆ 𝑐 → ((𝐹‘𝑏)𝑅(𝐹‘𝑐) ∨ (𝐹‘𝑏) = (𝐹‘𝑐))) ↔ (𝑑 ⊆ 𝑒 → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒)))))
25	17, 24	imbi12d 344	. . . . . . . . . . 11 ⊢ ((𝑏 = 𝑑 ∧ 𝑐 = 𝑒) → ((((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → (𝑏 ⊆ 𝑐 → ((𝐹‘𝑏)𝑅(𝐹‘𝑐) ∨ (𝐹‘𝑏) = (𝐹‘𝑐)))) ↔ (((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) ∧ (𝑑 ∈ ω ∧ 𝑒 ∈ ω)) → (𝑑 ⊆ 𝑒 → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒))))))
26		fveq2 6840	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑏 → (𝐹‘𝑑) = (𝐹‘𝑏))
27	26	breq2d 5114	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑏 → ((𝐹‘𝑏)𝑅(𝐹‘𝑑) ↔ (𝐹‘𝑏)𝑅(𝐹‘𝑏)))
28	26	eqeq2d 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑏 → ((𝐹‘𝑏) = (𝐹‘𝑑) ↔ (𝐹‘𝑏) = (𝐹‘𝑏)))
29	27, 28	orbi12d 918	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝑏 → (((𝐹‘𝑏)𝑅(𝐹‘𝑑) ∨ (𝐹‘𝑏) = (𝐹‘𝑑)) ↔ ((𝐹‘𝑏)𝑅(𝐹‘𝑏) ∨ (𝐹‘𝑏) = (𝐹‘𝑏))))
30	29	imbi2d 340	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑏 → (((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → ((𝐹‘𝑏)𝑅(𝐹‘𝑑) ∨ (𝐹‘𝑏) = (𝐹‘𝑑))) ↔ ((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → ((𝐹‘𝑏)𝑅(𝐹‘𝑏) ∨ (𝐹‘𝑏) = (𝐹‘𝑏)))))
31		fveq2 6840	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑒 → (𝐹‘𝑑) = (𝐹‘𝑒))
32	31	breq2d 5114	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑒 → ((𝐹‘𝑏)𝑅(𝐹‘𝑑) ↔ (𝐹‘𝑏)𝑅(𝐹‘𝑒)))
33	31	eqeq2d 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑒 → ((𝐹‘𝑏) = (𝐹‘𝑑) ↔ (𝐹‘𝑏) = (𝐹‘𝑒)))
34	32, 33	orbi12d 918	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝑒 → (((𝐹‘𝑏)𝑅(𝐹‘𝑑) ∨ (𝐹‘𝑏) = (𝐹‘𝑑)) ↔ ((𝐹‘𝑏)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑏) = (𝐹‘𝑒))))
35	34	imbi2d 340	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → (((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → ((𝐹‘𝑏)𝑅(𝐹‘𝑑) ∨ (𝐹‘𝑏) = (𝐹‘𝑑))) ↔ ((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → ((𝐹‘𝑏)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑏) = (𝐹‘𝑒)))))
36		fveq2 6840	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = suc 𝑒 → (𝐹‘𝑑) = (𝐹‘suc 𝑒))
37	36	breq2d 5114	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = suc 𝑒 → ((𝐹‘𝑏)𝑅(𝐹‘𝑑) ↔ (𝐹‘𝑏)𝑅(𝐹‘suc 𝑒)))
38	36	eqeq2d 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = suc 𝑒 → ((𝐹‘𝑏) = (𝐹‘𝑑) ↔ (𝐹‘𝑏) = (𝐹‘suc 𝑒)))
39	37, 38	orbi12d 918	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = suc 𝑒 → (((𝐹‘𝑏)𝑅(𝐹‘𝑑) ∨ (𝐹‘𝑏) = (𝐹‘𝑑)) ↔ ((𝐹‘𝑏)𝑅(𝐹‘suc 𝑒) ∨ (𝐹‘𝑏) = (𝐹‘suc 𝑒))))
40	39	imbi2d 340	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = suc 𝑒 → (((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → ((𝐹‘𝑏)𝑅(𝐹‘𝑑) ∨ (𝐹‘𝑏) = (𝐹‘𝑑))) ↔ ((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → ((𝐹‘𝑏)𝑅(𝐹‘suc 𝑒) ∨ (𝐹‘𝑏) = (𝐹‘suc 𝑒)))))
41		fveq2 6840	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑐 → (𝐹‘𝑑) = (𝐹‘𝑐))
42	41	breq2d 5114	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑐 → ((𝐹‘𝑏)𝑅(𝐹‘𝑑) ↔ (𝐹‘𝑏)𝑅(𝐹‘𝑐)))
43	41	eqeq2d 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑐 → ((𝐹‘𝑏) = (𝐹‘𝑑) ↔ (𝐹‘𝑏) = (𝐹‘𝑐)))
44	42, 43	orbi12d 918	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝑐 → (((𝐹‘𝑏)𝑅(𝐹‘𝑑) ∨ (𝐹‘𝑏) = (𝐹‘𝑑)) ↔ ((𝐹‘𝑏)𝑅(𝐹‘𝑐) ∨ (𝐹‘𝑏) = (𝐹‘𝑐))))
45	44	imbi2d 340	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑐 → (((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → ((𝐹‘𝑏)𝑅(𝐹‘𝑑) ∨ (𝐹‘𝑏) = (𝐹‘𝑑))) ↔ ((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → ((𝐹‘𝑏)𝑅(𝐹‘𝑐) ∨ (𝐹‘𝑏) = (𝐹‘𝑐)))))
46		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹‘𝑏) = (𝐹‘𝑏)
47	46	olci 866	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑏)𝑅(𝐹‘𝑏) ∨ (𝐹‘𝑏) = (𝐹‘𝑏))
48	47	2a1i 12	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ ω → ((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → ((𝐹‘𝑏)𝑅(𝐹‘𝑏) ∨ (𝐹‘𝑏) = (𝐹‘𝑏))))
49		fveq2 6840	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑒 → (𝐹‘𝑎) = (𝐹‘𝑒))
50		suceq 6388	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑒 → suc 𝑎 = suc 𝑒)
51	50	fveq2d 6844	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑒 → (𝐹‘suc 𝑎) = (𝐹‘suc 𝑒))
52	49, 51	breq12d 5115	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑒 → ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ↔ (𝐹‘𝑒)𝑅(𝐹‘suc 𝑒)))
53	49, 51	eqeq12d 2745	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑒 → ((𝐹‘𝑎) = (𝐹‘suc 𝑎) ↔ (𝐹‘𝑒) = (𝐹‘suc 𝑒)))
54	52, 53	orbi12d 918	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑒 → (((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ↔ ((𝐹‘𝑒)𝑅(𝐹‘suc 𝑒) ∨ (𝐹‘𝑒) = (𝐹‘suc 𝑒))))
55		simpr2 1196	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹)) → ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)))
56		simplll 774	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹)) → 𝑒 ∈ ω)
57	54, 55, 56	rspcdva 3586	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹)) → ((𝐹‘𝑒)𝑅(𝐹‘suc 𝑒) ∨ (𝐹‘𝑒) = (𝐹‘suc 𝑒)))
58		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → 𝑅 Po ran 𝐹)
59		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → 𝐹 Fn ω)
60		simpllr 775	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → 𝑏 ∈ ω)
61		fnfvelrn 7034	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐹 Fn ω ∧ 𝑏 ∈ ω) → (𝐹‘𝑏) ∈ ran 𝐹)
62	59, 60, 61	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → (𝐹‘𝑏) ∈ ran 𝐹)
63		simplll 774	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → 𝑒 ∈ ω)
64		fnfvelrn 7034	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐹 Fn ω ∧ 𝑒 ∈ ω) → (𝐹‘𝑒) ∈ ran 𝐹)
65	59, 63, 64	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → (𝐹‘𝑒) ∈ ran 𝐹)
66		peano2 7846	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑒 ∈ ω → suc 𝑒 ∈ ω)
67	66	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → suc 𝑒 ∈ ω)
68		fnfvelrn 7034	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐹 Fn ω ∧ suc 𝑒 ∈ ω) → (𝐹‘suc 𝑒) ∈ ran 𝐹)
69	59, 67, 68	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → (𝐹‘suc 𝑒) ∈ ran 𝐹)
70		potr 5552	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑅 Po ran 𝐹 ∧ ((𝐹‘𝑏) ∈ ran 𝐹 ∧ (𝐹‘𝑒) ∈ ran 𝐹 ∧ (𝐹‘suc 𝑒) ∈ ran 𝐹)) → (((𝐹‘𝑏)𝑅(𝐹‘𝑒) ∧ (𝐹‘𝑒)𝑅(𝐹‘suc 𝑒)) → (𝐹‘𝑏)𝑅(𝐹‘suc 𝑒)))
71	58, 62, 65, 69, 70	syl13anc 1374	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → (((𝐹‘𝑏)𝑅(𝐹‘𝑒) ∧ (𝐹‘𝑒)𝑅(𝐹‘suc 𝑒)) → (𝐹‘𝑏)𝑅(𝐹‘suc 𝑒)))
72	71	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) ∧ ((𝐹‘𝑏)𝑅(𝐹‘𝑒) ∧ (𝐹‘𝑒)𝑅(𝐹‘suc 𝑒))) → (𝐹‘𝑏)𝑅(𝐹‘suc 𝑒))
73	72	ancom2s 650	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) ∧ ((𝐹‘𝑒)𝑅(𝐹‘suc 𝑒) ∧ (𝐹‘𝑏)𝑅(𝐹‘𝑒))) → (𝐹‘𝑏)𝑅(𝐹‘suc 𝑒))
74	73	orcd 873	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) ∧ ((𝐹‘𝑒)𝑅(𝐹‘suc 𝑒) ∧ (𝐹‘𝑏)𝑅(𝐹‘𝑒))) → ((𝐹‘𝑏)𝑅(𝐹‘suc 𝑒) ∨ (𝐹‘𝑏) = (𝐹‘suc 𝑒)))
75	74	expr 456	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) ∧ (𝐹‘𝑒)𝑅(𝐹‘suc 𝑒)) → ((𝐹‘𝑏)𝑅(𝐹‘𝑒) → ((𝐹‘𝑏)𝑅(𝐹‘suc 𝑒) ∨ (𝐹‘𝑏) = (𝐹‘suc 𝑒))))
76		breq1 5105	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹‘𝑏) = (𝐹‘𝑒) → ((𝐹‘𝑏)𝑅(𝐹‘suc 𝑒) ↔ (𝐹‘𝑒)𝑅(𝐹‘suc 𝑒)))
77	76	biimprcd 250	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹‘𝑒)𝑅(𝐹‘suc 𝑒) → ((𝐹‘𝑏) = (𝐹‘𝑒) → (𝐹‘𝑏)𝑅(𝐹‘suc 𝑒)))
78		orc 867	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹‘𝑏)𝑅(𝐹‘suc 𝑒) → ((𝐹‘𝑏)𝑅(𝐹‘suc 𝑒) ∨ (𝐹‘𝑏) = (𝐹‘suc 𝑒)))
79	77, 78	syl6 35	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹‘𝑒)𝑅(𝐹‘suc 𝑒) → ((𝐹‘𝑏) = (𝐹‘𝑒) → ((𝐹‘𝑏)𝑅(𝐹‘suc 𝑒) ∨ (𝐹‘𝑏) = (𝐹‘suc 𝑒))))
80	79	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) ∧ (𝐹‘𝑒)𝑅(𝐹‘suc 𝑒)) → ((𝐹‘𝑏) = (𝐹‘𝑒) → ((𝐹‘𝑏)𝑅(𝐹‘suc 𝑒) ∨ (𝐹‘𝑏) = (𝐹‘suc 𝑒))))
81	75, 80	jaod 859	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) ∧ (𝐹‘𝑒)𝑅(𝐹‘suc 𝑒)) → (((𝐹‘𝑏)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑏) = (𝐹‘𝑒)) → ((𝐹‘𝑏)𝑅(𝐹‘suc 𝑒) ∨ (𝐹‘𝑏) = (𝐹‘suc 𝑒))))
82	81	ex 412	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → ((𝐹‘𝑒)𝑅(𝐹‘suc 𝑒) → (((𝐹‘𝑏)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑏) = (𝐹‘𝑒)) → ((𝐹‘𝑏)𝑅(𝐹‘suc 𝑒) ∨ (𝐹‘𝑏) = (𝐹‘suc 𝑒)))))
83		breq2 5106	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹‘𝑒) = (𝐹‘suc 𝑒) → ((𝐹‘𝑏)𝑅(𝐹‘𝑒) ↔ (𝐹‘𝑏)𝑅(𝐹‘suc 𝑒)))
84		eqeq2 2741	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹‘𝑒) = (𝐹‘suc 𝑒) → ((𝐹‘𝑏) = (𝐹‘𝑒) ↔ (𝐹‘𝑏) = (𝐹‘suc 𝑒)))
85	83, 84	orbi12d 918	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑒) = (𝐹‘suc 𝑒) → (((𝐹‘𝑏)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑏) = (𝐹‘𝑒)) ↔ ((𝐹‘𝑏)𝑅(𝐹‘suc 𝑒) ∨ (𝐹‘𝑏) = (𝐹‘suc 𝑒))))
86	85	biimpd 229	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘𝑒) = (𝐹‘suc 𝑒) → (((𝐹‘𝑏)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑏) = (𝐹‘𝑒)) → ((𝐹‘𝑏)𝑅(𝐹‘suc 𝑒) ∨ (𝐹‘𝑏) = (𝐹‘suc 𝑒))))
87	86	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → ((𝐹‘𝑒) = (𝐹‘suc 𝑒) → (((𝐹‘𝑏)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑏) = (𝐹‘𝑒)) → ((𝐹‘𝑏)𝑅(𝐹‘suc 𝑒) ∨ (𝐹‘𝑏) = (𝐹‘suc 𝑒)))))
88	82, 87	jaod 859	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → (((𝐹‘𝑒)𝑅(𝐹‘suc 𝑒) ∨ (𝐹‘𝑒) = (𝐹‘suc 𝑒)) → (((𝐹‘𝑏)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑏) = (𝐹‘𝑒)) → ((𝐹‘𝑏)𝑅(𝐹‘suc 𝑒) ∨ (𝐹‘𝑏) = (𝐹‘suc 𝑒)))))
89	88	3adantr2 1171	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹)) → (((𝐹‘𝑒)𝑅(𝐹‘suc 𝑒) ∨ (𝐹‘𝑒) = (𝐹‘suc 𝑒)) → (((𝐹‘𝑏)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑏) = (𝐹‘𝑒)) → ((𝐹‘𝑏)𝑅(𝐹‘suc 𝑒) ∨ (𝐹‘𝑏) = (𝐹‘suc 𝑒)))))
90	57, 89	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹)) → (((𝐹‘𝑏)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑏) = (𝐹‘𝑒)) → ((𝐹‘𝑏)𝑅(𝐹‘suc 𝑒) ∨ (𝐹‘𝑏) = (𝐹‘suc 𝑒))))
91	90	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) → ((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → (((𝐹‘𝑏)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑏) = (𝐹‘𝑒)) → ((𝐹‘𝑏)𝑅(𝐹‘suc 𝑒) ∨ (𝐹‘𝑏) = (𝐹‘suc 𝑒)))))
92	91	a2d 29	. . . . . . . . . . . . . . 15 ⊢ (((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) → (((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → ((𝐹‘𝑏)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑏) = (𝐹‘𝑒))) → ((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → ((𝐹‘𝑏)𝑅(𝐹‘suc 𝑒) ∨ (𝐹‘𝑏) = (𝐹‘suc 𝑒)))))
93	30, 35, 40, 45, 48, 92	findsg 7853	. . . . . . . . . . . . . 14 ⊢ (((𝑐 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑐) → ((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → ((𝐹‘𝑏)𝑅(𝐹‘𝑐) ∨ (𝐹‘𝑏) = (𝐹‘𝑐))))
94	93	ancom1s 653	. . . . . . . . . . . . 13 ⊢ (((𝑏 ∈ ω ∧ 𝑐 ∈ ω) ∧ 𝑏 ⊆ 𝑐) → ((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → ((𝐹‘𝑏)𝑅(𝐹‘𝑐) ∨ (𝐹‘𝑏) = (𝐹‘𝑐))))
95	94	impcom 407	. . . . . . . . . . . 12 ⊢ (((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) ∧ ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) ∧ 𝑏 ⊆ 𝑐)) → ((𝐹‘𝑏)𝑅(𝐹‘𝑐) ∨ (𝐹‘𝑏) = (𝐹‘𝑐)))
96	95	expr 456	. . . . . . . . . . 11 ⊢ (((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → (𝑏 ⊆ 𝑐 → ((𝐹‘𝑏)𝑅(𝐹‘𝑐) ∨ (𝐹‘𝑏) = (𝐹‘𝑐))))
97	12, 13, 25, 96	vtocl2 3529	. . . . . . . . . 10 ⊢ (((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) ∧ (𝑑 ∈ ω ∧ 𝑒 ∈ ω)) → (𝑑 ⊆ 𝑒 → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒))))
98		eleq1w 2811	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑒 → (𝑏 ∈ ω ↔ 𝑒 ∈ ω))
99		eleq1w 2811	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑑 → (𝑐 ∈ ω ↔ 𝑑 ∈ ω))
100	98, 99	bi2anan9 638	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 𝑒 ∧ 𝑐 = 𝑑) → ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) ↔ (𝑒 ∈ ω ∧ 𝑑 ∈ ω)))
101	100	anbi2d 630	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 𝑒 ∧ 𝑐 = 𝑑) → (((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ↔ ((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) ∧ (𝑒 ∈ ω ∧ 𝑑 ∈ ω))))
102		sseq12 3971	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 𝑒 ∧ 𝑐 = 𝑑) → (𝑏 ⊆ 𝑐 ↔ 𝑒 ⊆ 𝑑))
103		fveq2 6840	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑒 → (𝐹‘𝑏) = (𝐹‘𝑒))
104		fveq2 6840	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝑑 → (𝐹‘𝑐) = (𝐹‘𝑑))
105	103, 104	breqan12d 5118	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 𝑒 ∧ 𝑐 = 𝑑) → ((𝐹‘𝑏)𝑅(𝐹‘𝑐) ↔ (𝐹‘𝑒)𝑅(𝐹‘𝑑)))
106	103, 104	eqeqan12d 2743	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 𝑒 ∧ 𝑐 = 𝑑) → ((𝐹‘𝑏) = (𝐹‘𝑐) ↔ (𝐹‘𝑒) = (𝐹‘𝑑)))
107	105, 106	orbi12d 918	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 𝑒 ∧ 𝑐 = 𝑑) → (((𝐹‘𝑏)𝑅(𝐹‘𝑐) ∨ (𝐹‘𝑏) = (𝐹‘𝑐)) ↔ ((𝐹‘𝑒)𝑅(𝐹‘𝑑) ∨ (𝐹‘𝑒) = (𝐹‘𝑑))))
108	102, 107	imbi12d 344	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 𝑒 ∧ 𝑐 = 𝑑) → ((𝑏 ⊆ 𝑐 → ((𝐹‘𝑏)𝑅(𝐹‘𝑐) ∨ (𝐹‘𝑏) = (𝐹‘𝑐))) ↔ (𝑒 ⊆ 𝑑 → ((𝐹‘𝑒)𝑅(𝐹‘𝑑) ∨ (𝐹‘𝑒) = (𝐹‘𝑑)))))
109	101, 108	imbi12d 344	. . . . . . . . . . . 12 ⊢ ((𝑏 = 𝑒 ∧ 𝑐 = 𝑑) → ((((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → (𝑏 ⊆ 𝑐 → ((𝐹‘𝑏)𝑅(𝐹‘𝑐) ∨ (𝐹‘𝑏) = (𝐹‘𝑐)))) ↔ (((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) ∧ (𝑒 ∈ ω ∧ 𝑑 ∈ ω)) → (𝑒 ⊆ 𝑑 → ((𝐹‘𝑒)𝑅(𝐹‘𝑑) ∨ (𝐹‘𝑒) = (𝐹‘𝑑))))))
110	13, 12, 109, 96	vtocl2 3529	. . . . . . . . . . 11 ⊢ (((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) ∧ (𝑒 ∈ ω ∧ 𝑑 ∈ ω)) → (𝑒 ⊆ 𝑑 → ((𝐹‘𝑒)𝑅(𝐹‘𝑑) ∨ (𝐹‘𝑒) = (𝐹‘𝑑))))
111	110	ancom2s 650	. . . . . . . . . 10 ⊢ (((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) ∧ (𝑑 ∈ ω ∧ 𝑒 ∈ ω)) → (𝑒 ⊆ 𝑑 → ((𝐹‘𝑒)𝑅(𝐹‘𝑑) ∨ (𝐹‘𝑒) = (𝐹‘𝑑))))
112	97, 111	orim12d 966	. . . . . . . . 9 ⊢ (((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) ∧ (𝑑 ∈ ω ∧ 𝑒 ∈ ω)) → ((𝑑 ⊆ 𝑒 ∨ 𝑒 ⊆ 𝑑) → (((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒)) ∨ ((𝐹‘𝑒)𝑅(𝐹‘𝑑) ∨ (𝐹‘𝑒) = (𝐹‘𝑑)))))
113	11, 112	mpd 15	. . . . . . . 8 ⊢ (((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) ∧ (𝑑 ∈ ω ∧ 𝑒 ∈ ω)) → (((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒)) ∨ ((𝐹‘𝑒)𝑅(𝐹‘𝑑) ∨ (𝐹‘𝑒) = (𝐹‘𝑑))))
114		3mix1 1331	. . . . . . . . . 10 ⊢ ((𝐹‘𝑑)𝑅(𝐹‘𝑒) → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒) ∨ (𝐹‘𝑒)𝑅(𝐹‘𝑑)))
115		3mix2 1332	. . . . . . . . . 10 ⊢ ((𝐹‘𝑑) = (𝐹‘𝑒) → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒) ∨ (𝐹‘𝑒)𝑅(𝐹‘𝑑)))
116	114, 115	jaoi 857	. . . . . . . . 9 ⊢ (((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒)) → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒) ∨ (𝐹‘𝑒)𝑅(𝐹‘𝑑)))
117		3mix3 1333	. . . . . . . . . 10 ⊢ ((𝐹‘𝑒)𝑅(𝐹‘𝑑) → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒) ∨ (𝐹‘𝑒)𝑅(𝐹‘𝑑)))
118	115	eqcoms 2737	. . . . . . . . . 10 ⊢ ((𝐹‘𝑒) = (𝐹‘𝑑) → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒) ∨ (𝐹‘𝑒)𝑅(𝐹‘𝑑)))
119	117, 118	jaoi 857	. . . . . . . . 9 ⊢ (((𝐹‘𝑒)𝑅(𝐹‘𝑑) ∨ (𝐹‘𝑒) = (𝐹‘𝑑)) → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒) ∨ (𝐹‘𝑒)𝑅(𝐹‘𝑑)))
120	116, 119	jaoi 857	. . . . . . . 8 ⊢ ((((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒)) ∨ ((𝐹‘𝑒)𝑅(𝐹‘𝑑) ∨ (𝐹‘𝑒) = (𝐹‘𝑑))) → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒) ∨ (𝐹‘𝑒)𝑅(𝐹‘𝑑)))
121	113, 120	syl 17	. . . . . . 7 ⊢ (((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) ∧ (𝑑 ∈ ω ∧ 𝑒 ∈ ω)) → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒) ∨ (𝐹‘𝑒)𝑅(𝐹‘𝑑)))
122		breq12 5107	. . . . . . . 8 ⊢ (((𝐹‘𝑑) = 𝑏 ∧ (𝐹‘𝑒) = 𝑐) → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ↔ 𝑏𝑅𝑐))
123		eqeq12 2746	. . . . . . . 8 ⊢ (((𝐹‘𝑑) = 𝑏 ∧ (𝐹‘𝑒) = 𝑐) → ((𝐹‘𝑑) = (𝐹‘𝑒) ↔ 𝑏 = 𝑐))
124		breq12 5107	. . . . . . . . 9 ⊢ (((𝐹‘𝑒) = 𝑐 ∧ (𝐹‘𝑑) = 𝑏) → ((𝐹‘𝑒)𝑅(𝐹‘𝑑) ↔ 𝑐𝑅𝑏))
125	124	ancoms 458	. . . . . . . 8 ⊢ (((𝐹‘𝑑) = 𝑏 ∧ (𝐹‘𝑒) = 𝑐) → ((𝐹‘𝑒)𝑅(𝐹‘𝑑) ↔ 𝑐𝑅𝑏))
126	122, 123, 125	3orbi123d 1437	. . . . . . 7 ⊢ (((𝐹‘𝑑) = 𝑏 ∧ (𝐹‘𝑒) = 𝑐) → (((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒) ∨ (𝐹‘𝑒)𝑅(𝐹‘𝑑)) ↔ (𝑏𝑅𝑐 ∨ 𝑏 = 𝑐 ∨ 𝑐𝑅𝑏)))
127	121, 126	syl5ibcom 245	. . . . . 6 ⊢ (((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) ∧ (𝑑 ∈ ω ∧ 𝑒 ∈ ω)) → (((𝐹‘𝑑) = 𝑏 ∧ (𝐹‘𝑒) = 𝑐) → (𝑏𝑅𝑐 ∨ 𝑏 = 𝑐 ∨ 𝑐𝑅𝑏)))
128	127	rexlimdvva 3192	. . . . 5 ⊢ ((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → (∃𝑑 ∈ ω ∃𝑒 ∈ ω ((𝐹‘𝑑) = 𝑏 ∧ (𝐹‘𝑒) = 𝑐) → (𝑏𝑅𝑐 ∨ 𝑏 = 𝑐 ∨ 𝑐𝑅𝑏)))
129	6, 128	biimtrrid 243	. . . 4 ⊢ ((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → ((∃𝑑 ∈ ω (𝐹‘𝑑) = 𝑏 ∧ ∃𝑒 ∈ ω (𝐹‘𝑒) = 𝑐) → (𝑏𝑅𝑐 ∨ 𝑏 = 𝑐 ∨ 𝑐𝑅𝑏)))
130	5, 129	sylbid 240	. . 3 ⊢ ((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → ((𝑏 ∈ ran 𝐹 ∧ 𝑐 ∈ ran 𝐹) → (𝑏𝑅𝑐 ∨ 𝑏 = 𝑐 ∨ 𝑐𝑅𝑏)))
131	130	ralrimivv 3176	. 2 ⊢ ((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → ∀𝑏 ∈ ran 𝐹∀𝑐 ∈ ran 𝐹(𝑏𝑅𝑐 ∨ 𝑏 = 𝑐 ∨ 𝑐𝑅𝑏))
132		df-so 5540	. 2 ⊢ (𝑅 Or ran 𝐹 ↔ (𝑅 Po ran 𝐹 ∧ ∀𝑏 ∈ ran 𝐹∀𝑐 ∈ ran 𝐹(𝑏𝑅𝑐 ∨ 𝑏 = 𝑐 ∨ 𝑐𝑅𝑏)))
133	1, 131, 132	sylanbrc 583	1 ⊢ ((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → 𝑅 Or ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3911   class class class wbr 5102   Po wpo 5537   Or wor 5538  ran crn 5632  Ord word 6319  suc csuc 6322   Fn wfn 6494  ‘cfv 6499  ωcom 7822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-fv 6507  df-om 7823

This theorem is referenced by:  fin23lem40  10280