Theorem sornom 10237

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 6924	. . . . . 6 ⊢ (𝐹 Fn ω → (𝑏 ∈ ran 𝐹 ↔ ∃𝑑 ∈ ω (𝐹‘𝑑) = 𝑏))
3		fvelrnb 6924	. . . . . 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 3210	. . . . 5 ⊢ (∃𝑑 ∈ ω ∃𝑒 ∈ ω ((𝐹‘𝑑) = 𝑏 ∧ (𝐹‘𝑒) = 𝑐) ↔ (∃𝑑 ∈ ω (𝐹‘𝑑) = 𝑏 ∧ ∃𝑒 ∈ ω (𝐹‘𝑒) = 𝑐))
7		nnord 7853	. . . . . . . . . . 11 ⊢ (𝑑 ∈ ω → Ord 𝑑)
8		nnord 7853	. . . . . . . . . . 11 ⊢ (𝑒 ∈ ω → Ord 𝑒)
9		ordtri2or2 6436	. . . . . . . . . . 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 3454	. . . . . . . . . . 11 ⊢ 𝑑 ∈ V
13		vex 3454	. . . . . . . . . . 11 ⊢ 𝑒 ∈ V
14		eleq1w 2812	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑑 → (𝑏 ∈ ω ↔ 𝑑 ∈ ω))
15		eleq1w 2812	. . . . . . . . . . . . . 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 3977	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 𝑑 ∧ 𝑐 = 𝑒) → (𝑏 ⊆ 𝑐 ↔ 𝑑 ⊆ 𝑒))
19		fveq2 6861	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑑 → (𝐹‘𝑏) = (𝐹‘𝑑))
20		fveq2 6861	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑒 → (𝐹‘𝑐) = (𝐹‘𝑒))
21	19, 20	breqan12d 5126	. . . . . . . . . . . . . 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 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 6861	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑏 → (𝐹‘𝑑) = (𝐹‘𝑏))
27	26	breq2d 5122	. . . . . . . . . . . . . . . . 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 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 6861	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑒 → (𝐹‘𝑑) = (𝐹‘𝑒))
32	31	breq2d 5122	. . . . . . . . . . . . . . . . 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 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 6861	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = suc 𝑒 → (𝐹‘𝑑) = (𝐹‘suc 𝑒))
37	36	breq2d 5122	. . . . . . . . . . . . . . . . 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 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 6861	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑐 → (𝐹‘𝑑) = (𝐹‘𝑐))
42	41	breq2d 5122	. . . . . . . . . . . . . . . . 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 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 2730	. . . . . . . . . . . . . . . . 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 6861	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑒 → (𝐹‘𝑎) = (𝐹‘𝑒))
50		suceq 6403	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑒 → suc 𝑎 = suc 𝑒)
51	50	fveq2d 6865	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑒 → (𝐹‘suc 𝑎) = (𝐹‘suc 𝑒))
52	49, 51	breq12d 5123	. . . . . . . . . . . . . . . . . . . 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 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 3592	. . . . . . . . . . . . . . . . . 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 7055	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7055	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7869	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7055	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5562	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 5113	. . . . . . . . . . . . . . . . . . . . . . . . 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 5114	. . . . . . . . . . . . . . . . . . . . . . 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 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 7876	. . . . . . . . . . . . . 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 3535	. . . . . . . . . 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 2812	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑒 → (𝑏 ∈ ω ↔ 𝑒 ∈ ω))
99		eleq1w 2812	. . . . . . . . . . . . . . 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 3977	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 𝑒 ∧ 𝑐 = 𝑑) → (𝑏 ⊆ 𝑐 ↔ 𝑒 ⊆ 𝑑))
103		fveq2 6861	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑒 → (𝐹‘𝑏) = (𝐹‘𝑒))
104		fveq2 6861	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝑑 → (𝐹‘𝑐) = (𝐹‘𝑑))
105	103, 104	breqan12d 5126	. . . . . . . . . . . . . . 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 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 3535	. . . . . . . . . . 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 2738	. . . . . . . . . 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 5115	. . . . . . . 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 5115	. . . . . . . . 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 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 5550	. 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 3045  ∃wrex 3054   ⊆ wss 3917   class class class wbr 5110   Po wpo 5547   Or wor 5548  ran crn 5642  Ord word 6334  suc csuc 6337   Fn wfn 6509  ‘cfv 6514  ωcom 7845

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-fv 6522  df-om 7846

This theorem is referenced by:  fin23lem40  10311