Theorem sornom 10276

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 1136	. 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 6953	. . . . . 6 ⊢ (𝐹 Fn ω → (𝑏 ∈ ran 𝐹 ↔ ∃𝑑 ∈ ω (𝐹‘𝑑) = 𝑏))
3		fvelrnb 6953	. . . . . 6 ⊢ (𝐹 Fn ω → (𝑐 ∈ ran 𝐹 ↔ ∃𝑒 ∈ ω (𝐹‘𝑒) = 𝑐))
4	2, 3	anbi12d 629	. . . . 5 ⊢ (𝐹 Fn ω → ((𝑏 ∈ ran 𝐹 ∧ 𝑐 ∈ ran 𝐹) ↔ (∃𝑑 ∈ ω (𝐹‘𝑑) = 𝑏 ∧ ∃𝑒 ∈ ω (𝐹‘𝑒) = 𝑐)))
5	4	3ad2ant1 1131	. . . 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 3224	. . . . 5 ⊢ (∃𝑑 ∈ ω ∃𝑒 ∈ ω ((𝐹‘𝑑) = 𝑏 ∧ (𝐹‘𝑒) = 𝑐) ↔ (∃𝑑 ∈ ω (𝐹‘𝑑) = 𝑏 ∧ ∃𝑒 ∈ ω (𝐹‘𝑒) = 𝑐))
7		nnord 7867	. . . . . . . . . . 11 ⊢ (𝑑 ∈ ω → Ord 𝑑)
8		nnord 7867	. . . . . . . . . . 11 ⊢ (𝑒 ∈ ω → Ord 𝑒)
9		ordtri2or2 6464	. . . . . . . . . . 11 ⊢ ((Ord 𝑑 ∧ Ord 𝑒) → (𝑑 ⊆ 𝑒 ∨ 𝑒 ⊆ 𝑑))
10	7, 8, 9	syl2an 594	. . . . . . . . . 10 ⊢ ((𝑑 ∈ ω ∧ 𝑒 ∈ ω) → (𝑑 ⊆ 𝑒 ∨ 𝑒 ⊆ 𝑑))
11	10	adantl 480	. . . . . . . . 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 3476	. . . . . . . . . . 11 ⊢ 𝑑 ∈ V
13		vex 3476	. . . . . . . . . . 11 ⊢ 𝑒 ∈ V
14		eleq1w 2814	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑑 → (𝑏 ∈ ω ↔ 𝑑 ∈ ω))
15		eleq1w 2814	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑒 → (𝑐 ∈ ω ↔ 𝑒 ∈ ω))
16	14, 15	bi2anan9 635	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 𝑑 ∧ 𝑐 = 𝑒) → ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) ↔ (𝑑 ∈ ω ∧ 𝑒 ∈ ω)))
17	16	anbi2d 627	. . . . . . . . . . . 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 4010	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 𝑑 ∧ 𝑐 = 𝑒) → (𝑏 ⊆ 𝑐 ↔ 𝑑 ⊆ 𝑒))
19		fveq2 6892	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑑 → (𝐹‘𝑏) = (𝐹‘𝑑))
20		fveq2 6892	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑒 → (𝐹‘𝑐) = (𝐹‘𝑒))
21	19, 20	breqan12d 5165	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 𝑑 ∧ 𝑐 = 𝑒) → ((𝐹‘𝑏)𝑅(𝐹‘𝑐) ↔ (𝐹‘𝑑)𝑅(𝐹‘𝑒)))
22	19, 20	eqeqan12d 2744	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 𝑑 ∧ 𝑐 = 𝑒) → ((𝐹‘𝑏) = (𝐹‘𝑐) ↔ (𝐹‘𝑑) = (𝐹‘𝑒)))
23	21, 22	orbi12d 915	. . . . . . . . . . . . 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 343	. . . . . . . . . . . 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 343	. . . . . . . . . . 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 6892	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑏 → (𝐹‘𝑑) = (𝐹‘𝑏))
27	26	breq2d 5161	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑏 → ((𝐹‘𝑏)𝑅(𝐹‘𝑑) ↔ (𝐹‘𝑏)𝑅(𝐹‘𝑏)))
28	26	eqeq2d 2741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑏 → ((𝐹‘𝑏) = (𝐹‘𝑑) ↔ (𝐹‘𝑏) = (𝐹‘𝑏)))
29	27, 28	orbi12d 915	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝑏 → (((𝐹‘𝑏)𝑅(𝐹‘𝑑) ∨ (𝐹‘𝑏) = (𝐹‘𝑑)) ↔ ((𝐹‘𝑏)𝑅(𝐹‘𝑏) ∨ (𝐹‘𝑏) = (𝐹‘𝑏))))
30	29	imbi2d 339	. . . . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑒 → (𝐹‘𝑑) = (𝐹‘𝑒))
32	31	breq2d 5161	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑒 → ((𝐹‘𝑏)𝑅(𝐹‘𝑑) ↔ (𝐹‘𝑏)𝑅(𝐹‘𝑒)))
33	31	eqeq2d 2741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑒 → ((𝐹‘𝑏) = (𝐹‘𝑑) ↔ (𝐹‘𝑏) = (𝐹‘𝑒)))
34	32, 33	orbi12d 915	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝑒 → (((𝐹‘𝑏)𝑅(𝐹‘𝑑) ∨ (𝐹‘𝑏) = (𝐹‘𝑑)) ↔ ((𝐹‘𝑏)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑏) = (𝐹‘𝑒))))
35	34	imbi2d 339	. . . . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = suc 𝑒 → (𝐹‘𝑑) = (𝐹‘suc 𝑒))
37	36	breq2d 5161	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = suc 𝑒 → ((𝐹‘𝑏)𝑅(𝐹‘𝑑) ↔ (𝐹‘𝑏)𝑅(𝐹‘suc 𝑒)))
38	36	eqeq2d 2741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = suc 𝑒 → ((𝐹‘𝑏) = (𝐹‘𝑑) ↔ (𝐹‘𝑏) = (𝐹‘suc 𝑒)))
39	37, 38	orbi12d 915	. . . . . . . . . . . . . . . 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 339	. . . . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑐 → (𝐹‘𝑑) = (𝐹‘𝑐))
42	41	breq2d 5161	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑐 → ((𝐹‘𝑏)𝑅(𝐹‘𝑑) ↔ (𝐹‘𝑏)𝑅(𝐹‘𝑐)))
43	41	eqeq2d 2741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑐 → ((𝐹‘𝑏) = (𝐹‘𝑑) ↔ (𝐹‘𝑏) = (𝐹‘𝑐)))
44	42, 43	orbi12d 915	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝑐 → (((𝐹‘𝑏)𝑅(𝐹‘𝑑) ∨ (𝐹‘𝑏) = (𝐹‘𝑑)) ↔ ((𝐹‘𝑏)𝑅(𝐹‘𝑐) ∨ (𝐹‘𝑏) = (𝐹‘𝑐))))
45	44	imbi2d 339	. . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹‘𝑏) = (𝐹‘𝑏)
47	46	olci 862	. . . . . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑒 → (𝐹‘𝑎) = (𝐹‘𝑒))
50		suceq 6431	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑒 → suc 𝑎 = suc 𝑒)
51	50	fveq2d 6896	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑒 → (𝐹‘suc 𝑎) = (𝐹‘suc 𝑒))
52	49, 51	breq12d 5162	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑒 → ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ↔ (𝐹‘𝑒)𝑅(𝐹‘suc 𝑒)))
53	49, 51	eqeq12d 2746	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑒 → ((𝐹‘𝑎) = (𝐹‘suc 𝑎) ↔ (𝐹‘𝑒) = (𝐹‘suc 𝑒)))
54	52, 53	orbi12d 915	. . . . . . . . . . . . . . . . . . 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 1193	. . . . . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . . . 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 3614	. . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → 𝑅 Po ran 𝐹)
59		simprl 767	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → 𝐹 Fn ω)
60		simpllr 772	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → 𝑏 ∈ ω)
61		fnfvelrn 7083	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐹 Fn ω ∧ 𝑏 ∈ ω) → (𝐹‘𝑏) ∈ ran 𝐹)
62	59, 60, 61	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → (𝐹‘𝑏) ∈ ran 𝐹)
63		simplll 771	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → 𝑒 ∈ ω)
64		fnfvelrn 7083	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐹 Fn ω ∧ 𝑒 ∈ ω) → (𝐹‘𝑒) ∈ ran 𝐹)
65	59, 63, 64	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → (𝐹‘𝑒) ∈ ran 𝐹)
66		peano2 7885	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑒 ∈ ω → suc 𝑒 ∈ ω)
67	66	ad3antrrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → suc 𝑒 ∈ ω)
68		fnfvelrn 7083	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐹 Fn ω ∧ suc 𝑒 ∈ ω) → (𝐹‘suc 𝑒) ∈ ran 𝐹)
69	59, 67, 68	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑒 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑒) ∧ (𝐹 Fn ω ∧ 𝑅 Po ran 𝐹)) → (𝐹‘suc 𝑒) ∈ ran 𝐹)
70		potr 5602	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . . . . . . . . . . . . . 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 646	. . . . . . . . . . . . . . . . . . . . . . . 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 869	. . . . . . . . . . . . . . . . . . . . . . 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 455	. . . . . . . . . . . . . . . . . . . . . 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 5152	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹‘𝑏) = (𝐹‘𝑒) → ((𝐹‘𝑏)𝑅(𝐹‘suc 𝑒) ↔ (𝐹‘𝑒)𝑅(𝐹‘suc 𝑒)))
77	76	biimprcd 249	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹‘𝑒)𝑅(𝐹‘suc 𝑒) → ((𝐹‘𝑏) = (𝐹‘𝑒) → (𝐹‘𝑏)𝑅(𝐹‘suc 𝑒)))
78		orc 863	. . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . 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 855	. . . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . . . 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 5153	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹‘𝑒) = (𝐹‘suc 𝑒) → ((𝐹‘𝑏)𝑅(𝐹‘𝑒) ↔ (𝐹‘𝑏)𝑅(𝐹‘suc 𝑒)))
84		eqeq2 2742	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹‘𝑒) = (𝐹‘suc 𝑒) → ((𝐹‘𝑏) = (𝐹‘𝑒) ↔ (𝐹‘𝑏) = (𝐹‘suc 𝑒)))
85	83, 84	orbi12d 915	. . . . . . . . . . . . . . . . . . . . . 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 228	. . . . . . . . . . . . . . . . . . . . 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 855	. . . . . . . . . . . . . . . . . . 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 1168	. . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . 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 7894	. . . . . . . . . . . . . 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 649	. . . . . . . . . . . . 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 406	. . . . . . . . . . . 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 455	. . . . . . . . . . 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 3553	. . . . . . . . . 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 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑒 → (𝑏 ∈ ω ↔ 𝑒 ∈ ω))
99		eleq1w 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑑 → (𝑐 ∈ ω ↔ 𝑑 ∈ ω))
100	98, 99	bi2anan9 635	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 𝑒 ∧ 𝑐 = 𝑑) → ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) ↔ (𝑒 ∈ ω ∧ 𝑑 ∈ ω)))
101	100	anbi2d 627	. . . . . . . . . . . . 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 4010	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 𝑒 ∧ 𝑐 = 𝑑) → (𝑏 ⊆ 𝑐 ↔ 𝑒 ⊆ 𝑑))
103		fveq2 6892	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑒 → (𝐹‘𝑏) = (𝐹‘𝑒))
104		fveq2 6892	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝑑 → (𝐹‘𝑐) = (𝐹‘𝑑))
105	103, 104	breqan12d 5165	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 𝑒 ∧ 𝑐 = 𝑑) → ((𝐹‘𝑏)𝑅(𝐹‘𝑐) ↔ (𝐹‘𝑒)𝑅(𝐹‘𝑑)))
106	103, 104	eqeqan12d 2744	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 𝑒 ∧ 𝑐 = 𝑑) → ((𝐹‘𝑏) = (𝐹‘𝑐) ↔ (𝐹‘𝑒) = (𝐹‘𝑑)))
107	105, 106	orbi12d 915	. . . . . . . . . . . . . 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 343	. . . . . . . . . . . . 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 343	. . . . . . . . . . . 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 3553	. . . . . . . . . . 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 646	. . . . . . . . . 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 961	. . . . . . . . 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 1328	. . . . . . . . . 10 ⊢ ((𝐹‘𝑑)𝑅(𝐹‘𝑒) → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒) ∨ (𝐹‘𝑒)𝑅(𝐹‘𝑑)))
115		3mix2 1329	. . . . . . . . . 10 ⊢ ((𝐹‘𝑑) = (𝐹‘𝑒) → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒) ∨ (𝐹‘𝑒)𝑅(𝐹‘𝑑)))
116	114, 115	jaoi 853	. . . . . . . . 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 1330	. . . . . . . . . 10 ⊢ ((𝐹‘𝑒)𝑅(𝐹‘𝑑) → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒) ∨ (𝐹‘𝑒)𝑅(𝐹‘𝑑)))
118	115	eqcoms 2738	. . . . . . . . . 10 ⊢ ((𝐹‘𝑒) = (𝐹‘𝑑) → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ∨ (𝐹‘𝑑) = (𝐹‘𝑒) ∨ (𝐹‘𝑒)𝑅(𝐹‘𝑑)))
119	117, 118	jaoi 853	. . . . . . . . 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 853	. . . . . . . 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 5154	. . . . . . . 8 ⊢ (((𝐹‘𝑑) = 𝑏 ∧ (𝐹‘𝑒) = 𝑐) → ((𝐹‘𝑑)𝑅(𝐹‘𝑒) ↔ 𝑏𝑅𝑐))
123		eqeq12 2747	. . . . . . . 8 ⊢ (((𝐹‘𝑑) = 𝑏 ∧ (𝐹‘𝑒) = 𝑐) → ((𝐹‘𝑑) = (𝐹‘𝑒) ↔ 𝑏 = 𝑐))
124		breq12 5154	. . . . . . . . 9 ⊢ (((𝐹‘𝑒) = 𝑐 ∧ (𝐹‘𝑑) = 𝑏) → ((𝐹‘𝑒)𝑅(𝐹‘𝑑) ↔ 𝑐𝑅𝑏))
125	124	ancoms 457	. . . . . . . 8 ⊢ (((𝐹‘𝑑) = 𝑏 ∧ (𝐹‘𝑒) = 𝑐) → ((𝐹‘𝑒)𝑅(𝐹‘𝑑) ↔ 𝑐𝑅𝑏))
126	122, 123, 125	3orbi123d 1433	. . . . . . 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 244	. . . . . 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 3209	. . . . 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 242	. . . 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 239	. . 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 3196	. 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 5590	. 2 ⊢ (𝑅 Or ran 𝐹 ↔ (𝑅 Po ran 𝐹 ∧ ∀𝑏 ∈ ran 𝐹∀𝑐 ∈ ran 𝐹(𝑏𝑅𝑐 ∨ 𝑏 = 𝑐 ∨ 𝑐𝑅𝑏)))
133	1, 131, 132	sylanbrc 581	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 205   ∧ wa 394   ∨ wo 843   ∨ w3o 1084   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  ∀wral 3059  ∃wrex 3068   ⊆ wss 3949   class class class wbr 5149   Po wpo 5587   Or wor 5588  ran crn 5678  Ord word 6364  suc csuc 6367   Fn wfn 6539  ‘cfv 6544  ωcom 7859

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-om 7860

This theorem is referenced by:  fin23lem40  10350