Theorem sornom 10201

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 1139	. 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 6904	. . . . . 6 ⊢ (𝐹 Fn ω → (𝑏 ∈ ran 𝐹 ↔ ∃𝑑 ∈ ω (𝐹‘𝑑) = 𝑏))
3		fvelrnb 6904	. . . . . 6 ⊢ (𝐹 Fn ω → (𝑐 ∈ ran 𝐹 ↔ ∃𝑒 ∈ ω (𝐹‘𝑒) = 𝑐))
4	2, 3	anbi12d 633	. . . . 5 ⊢ (𝐹 Fn ω → ((𝑏 ∈ ran 𝐹 ∧ 𝑐 ∈ ran 𝐹) ↔ (∃𝑑 ∈ ω (𝐹‘𝑑) = 𝑏 ∧ ∃𝑒 ∈ ω (𝐹‘𝑒) = 𝑐)))
5	4	3ad2ant1 1134	. . . 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 3210	. . . . 5 ⊢ (∃𝑑 ∈ ω ∃𝑒 ∈ ω ((𝐹‘𝑑) = 𝑏 ∧ (𝐹‘𝑒) = 𝑐) ↔ (∃𝑑 ∈ ω (𝐹‘𝑑) = 𝑏 ∧ ∃𝑒 ∈ ω (𝐹‘𝑒) = 𝑐))
7		nnord 7828	. . . . . . . . . . 11 ⊢ (𝑑 ∈ ω → Ord 𝑑)
8		nnord 7828	. . . . . . . . . . 11 ⊢ (𝑒 ∈ ω → Ord 𝑒)
9		ordtri2or2 6428	. . . . . . . . . . 11 ⊢ ((Ord 𝑑 ∧ Ord 𝑒) → (𝑑 ⊆ 𝑒 ∨ 𝑒 ⊆ 𝑑))
10	7, 8, 9	syl2an 597	. . . . . . . . . 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 3446	. . . . . . . . . . 11 ⊢ 𝑑 ∈ V
13		vex 3446	. . . . . . . . . . 11 ⊢ 𝑒 ∈ V
14		eleq1w 2820	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑑 → (𝑏 ∈ ω ↔ 𝑑 ∈ ω))
15		eleq1w 2820	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑒 → (𝑐 ∈ ω ↔ 𝑒 ∈ ω))
16	14, 15	bi2anan9 639	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 𝑑 ∧ 𝑐 = 𝑒) → ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) ↔ (𝑑 ∈ ω ∧ 𝑒 ∈ ω)))
17	16	anbi2d 631	. . . . . . . . . . . 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 3963	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 𝑑 ∧ 𝑐 = 𝑒) → (𝑏 ⊆ 𝑐 ↔ 𝑑 ⊆ 𝑒))
19		fveq2 6844	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑑 → (𝐹‘𝑏) = (𝐹‘𝑑))
20		fveq2 6844	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑒 → (𝐹‘𝑐) = (𝐹‘𝑒))
21	19, 20	breqan12d 5116	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 𝑑 ∧ 𝑐 = 𝑒) → ((𝐹‘𝑏)𝑅(𝐹‘𝑐) ↔ (𝐹‘𝑑)𝑅(𝐹‘𝑒)))
22	19, 20	eqeqan12d 2751	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 𝑑 ∧ 𝑐 = 𝑒) → ((𝐹‘𝑏) = (𝐹‘𝑐) ↔ (𝐹‘𝑑) = (𝐹‘𝑒)))
23	21, 22	orbi12d 919	. . . . . . . . . . . . 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 6844	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑏 → (𝐹‘𝑑) = (𝐹‘𝑏))
27	26	breq2d 5112	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑏 → ((𝐹‘𝑏)𝑅(𝐹‘𝑑) ↔ (𝐹‘𝑏)𝑅(𝐹‘𝑏)))
28	26	eqeq2d 2748	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑏 → ((𝐹‘𝑏) = (𝐹‘𝑑) ↔ (𝐹‘𝑏) = (𝐹‘𝑏)))
29	27, 28	orbi12d 919	. . . . . . . . . . . . . . . 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 6844	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑒 → (𝐹‘𝑑) = (𝐹‘𝑒))
32	31	breq2d 5112	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑒 → ((𝐹‘𝑏)𝑅(𝐹‘𝑑) ↔ (𝐹‘𝑏)𝑅(𝐹‘𝑒)))
33	31	eqeq2d 2748	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑒 → ((𝐹‘𝑏) = (𝐹‘𝑑) ↔ (𝐹‘𝑏) = (𝐹‘𝑒)))
34	32, 33	orbi12d 919	. . . . . . . . . . . . . . . 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 6844	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = suc 𝑒 → (𝐹‘𝑑) = (𝐹‘suc 𝑒))
37	36	breq2d 5112	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = suc 𝑒 → ((𝐹‘𝑏)𝑅(𝐹‘𝑑) ↔ (𝐹‘𝑏)𝑅(𝐹‘suc 𝑒)))
38	36	eqeq2d 2748	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = suc 𝑒 → ((𝐹‘𝑏) = (𝐹‘𝑑) ↔ (𝐹‘𝑏) = (𝐹‘suc 𝑒)))
39	37, 38	orbi12d 919	. . . . . . . . . . . . . . . 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 6844	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑐 → (𝐹‘𝑑) = (𝐹‘𝑐))
42	41	breq2d 5112	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑐 → ((𝐹‘𝑏)𝑅(𝐹‘𝑑) ↔ (𝐹‘𝑏)𝑅(𝐹‘𝑐)))
43	41	eqeq2d 2748	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑐 → ((𝐹‘𝑏) = (𝐹‘𝑑) ↔ (𝐹‘𝑏) = (𝐹‘𝑐)))
44	42, 43	orbi12d 919	. . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹‘𝑏) = (𝐹‘𝑏)
47	46	olci 867	. . . . . . . . . . . . . . . 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 6844	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑒 → (𝐹‘𝑎) = (𝐹‘𝑒))
50		suceq 6395	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑒 → suc 𝑎 = suc 𝑒)
51	50	fveq2d 6848	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑒 → (𝐹‘suc 𝑎) = (𝐹‘suc 𝑒))
52	49, 51	breq12d 5113	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑒 → ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ↔ (𝐹‘𝑒)𝑅(𝐹‘suc 𝑒)))
53	49, 51	eqeq12d 2753	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑒 → ((𝐹‘𝑎) = (𝐹‘suc 𝑎) ↔ (𝐹‘𝑒) = (𝐹‘suc 𝑒)))
54	52, 53	orbi12d 919	. . . . . . . . . . . . . . . . . . 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 1197	. . . . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . . 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 3579	. . . . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → 𝑅 Po ran 𝐹)
59		simprl 771	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → 𝐹 Fn ω)
60		simpllr 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → 𝑏 ∈ ω)
61		fnfvelrn 7036	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐹 Fn ω ∧ 𝑏 ∈ ω) → (𝐹‘𝑏) ∈ ran 𝐹)
62	59, 60, 61	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → (𝐹‘𝑏) ∈ ran 𝐹)
63		simplll 775	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → 𝑒 ∈ ω)
64		fnfvelrn 7036	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐹 Fn ω ∧ 𝑒 ∈ ω) → (𝐹‘𝑒) ∈ ran 𝐹)
65	59, 63, 64	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → (𝐹‘𝑒) ∈ ran 𝐹)
66		peano2 7844	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑒 ∈ ω → suc 𝑒 ∈ ω)
67	66	ad3antrrr 731	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → suc 𝑒 ∈ ω)
68		fnfvelrn 7036	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐹 Fn ω ∧ suc 𝑒 ∈ ω) → (𝐹‘suc 𝑒) ∈ ran 𝐹)
69	59, 67, 68	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → (𝐹‘suc 𝑒) ∈ ran 𝐹)
70		potr 5555	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . . . . . . . . . . . . . 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 651	. . . . . . . . . . . . . . . . . . . . . . . 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 874	. . . . . . . . . . . . . . . . . . . . . . 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 5103	. . . . . . . . . . . . . . . . . . . . . . . . 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 868	. . . . . . . . . . . . . . . . . . . . . . . 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 860	. . . . . . . . . . . . . . . . . . . . 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 5104	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹‘𝑒) = (𝐹‘suc 𝑒) → ((𝐹‘𝑏)𝑅(𝐹‘𝑒) ↔ (𝐹‘𝑏)𝑅(𝐹‘suc 𝑒)))
84		eqeq2 2749	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹‘𝑒) = (𝐹‘suc 𝑒) → ((𝐹‘𝑏) = (𝐹‘𝑒) ↔ (𝐹‘𝑏) = (𝐹‘suc 𝑒)))
85	83, 84	orbi12d 919	. . . . . . . . . . . . . . . . . . . . . 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 860	. . . . . . . . . . . . . . . . . . 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 1172	. . . . . . . . . . . . . . . . . 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 7851	. . . . . . . . . . . . . 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 654	. . . . . . . . . . . . 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 3524	. . . . . . . . . 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 2820	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑒 → (𝑏 ∈ ω ↔ 𝑒 ∈ ω))
99		eleq1w 2820	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑑 → (𝑐 ∈ ω ↔ 𝑑 ∈ ω))
100	98, 99	bi2anan9 639	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 𝑒 ∧ 𝑐 = 𝑑) → ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) ↔ (𝑒 ∈ ω ∧ 𝑑 ∈ ω)))
101	100	anbi2d 631	. . . . . . . . . . . . 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 3963	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 𝑒 ∧ 𝑐 = 𝑑) → (𝑏 ⊆ 𝑐 ↔ 𝑒 ⊆ 𝑑))
103		fveq2 6844	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑒 → (𝐹‘𝑏) = (𝐹‘𝑒))
104		fveq2 6844	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝑑 → (𝐹‘𝑐) = (𝐹‘𝑑))
105	103, 104	breqan12d 5116	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 𝑒 ∧ 𝑐 = 𝑑) → ((𝐹‘𝑏)𝑅(𝐹‘𝑐) ↔ (𝐹‘𝑒)𝑅(𝐹‘𝑑)))
106	103, 104	eqeqan12d 2751	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 𝑒 ∧ 𝑐 = 𝑑) → ((𝐹‘𝑏) = (𝐹‘𝑐) ↔ (𝐹‘𝑒) = (𝐹‘𝑑)))
107	105, 106	orbi12d 919	. . . . . . . . . . . . . 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 3524	. . . . . . . . . . 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 651	. . . . . . . . . 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 967	. . . . . . . . 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 1332	. . . . . . . . . 10 ⊢ ((𝐹‘𝑑)𝑅(𝐹‘𝑒) → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒) ∨ (𝐹‘𝑒)𝑅(𝐹‘𝑑)))
115		3mix2 1333	. . . . . . . . . 10 ⊢ ((𝐹‘𝑑) = (𝐹‘𝑒) → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒) ∨ (𝐹‘𝑒)𝑅(𝐹‘𝑑)))
116	114, 115	jaoi 858	. . . . . . . . 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 1334	. . . . . . . . . 10 ⊢ ((𝐹‘𝑒)𝑅(𝐹‘𝑑) → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒) ∨ (𝐹‘𝑒)𝑅(𝐹‘𝑑)))
118	115	eqcoms 2745	. . . . . . . . . 10 ⊢ ((𝐹‘𝑒) = (𝐹‘𝑑) → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒) ∨ (𝐹‘𝑒)𝑅(𝐹‘𝑑)))
119	117, 118	jaoi 858	. . . . . . . . 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 858	. . . . . . . 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 5105	. . . . . . . 8 ⊢ (((𝐹‘𝑑) = 𝑏 ∧ (𝐹‘𝑒) = 𝑐) → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ↔ 𝑏𝑅𝑐))
123		eqeq12 2754	. . . . . . . 8 ⊢ (((𝐹‘𝑑) = 𝑏 ∧ (𝐹‘𝑒) = 𝑐) → ((𝐹‘𝑑) = (𝐹‘𝑒) ↔ 𝑏 = 𝑐))
124		breq12 5105	. . . . . . . . 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 1438	. . . . . . 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 3195	. . . . 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 3179	. 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 5543	. 2 ⊢ (𝑅 Or ran 𝐹 ↔ (𝑅 Po ran 𝐹 ∧ ∀𝑏 ∈ ran 𝐹∀𝑐 ∈ ran 𝐹(𝑏𝑅𝑐 ∨ 𝑏 = 𝑐 ∨ 𝑐𝑅𝑏)))
133	1, 131, 132	sylanbrc 584	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 848   ∨ w3o 1086   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3903   class class class wbr 5100   Po wpo 5540   Or wor 5541  ran crn 5635  Ord word 6326  suc csuc 6329   Fn wfn 6497  ‘cfv 6502  ωcom 7820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-fv 6510  df-om 7821

This theorem is referenced by:  fin23lem40  10275