Theorem ordtypelem7 9521

Description: Lemma for ordtype 9529. ran 𝑂 is an initial segment of 𝐴 under the well-order 𝑅. (Contributed by Mario Carneiro, 25-Jun-2015.)

Hypotheses

Ref	Expression
ordtypelem.1	⊢ 𝐹 = recs(𝐺)
ordtypelem.2	⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
ordtypelem.3	⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))
ordtypelem.5	⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}
ordtypelem.6	⊢ 𝑂 = OrdIso(𝑅, 𝐴)
ordtypelem.7	⊢ (𝜑 → 𝑅 We 𝐴)
ordtypelem.8	⊢ (𝜑 → 𝑅 Se 𝐴)

Assertion

Ref	Expression
ordtypelem7	⊢ (((𝜑 ∧ 𝑁 ∈ 𝐴) ∧ 𝑀 ∈ dom 𝑂) → ((𝑂‘𝑀)𝑅𝑁 ∨ 𝑁 ∈ ran 𝑂))

Distinct variable groups:   𝑣,𝑢,𝐶   ℎ,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧,𝑀   𝑗,𝑁,𝑢,𝑤   𝑅,ℎ,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧   𝐴,ℎ,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧   𝑡,𝑂,𝑢,𝑣,𝑥   𝜑,𝑡,𝑥   ℎ,𝐹,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧

Allowed substitution hints:   𝜑(𝑧, 𝑤, 𝑣, 𝑢, ℎ, 𝑗)   𝐶(𝑥, 𝑧, 𝑤, 𝑡, ℎ, 𝑗)   𝑇(𝑥, 𝑧, 𝑤, 𝑣, 𝑢, 𝑡, ℎ, 𝑗)   𝐺(𝑥, 𝑧, 𝑤, 𝑣, 𝑢, 𝑡, ℎ, 𝑗)   𝑁(𝑥, 𝑧, 𝑣, 𝑡, ℎ)   𝑂(𝑧, 𝑤, ℎ, 𝑗)

Proof of Theorem ordtypelem7

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldif 4049	. . . . . 6 ⊢ (𝑁 ∈ (𝐴 ∖ ran 𝑂) ↔ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ ran 𝑂))
2		ordtypelem.1	. . . . . . . . . . . 12 ⊢ 𝐹 = recs(𝐺)
3		ordtypelem.2	. . . . . . . . . . . 12 ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
4		ordtypelem.3	. . . . . . . . . . . 12 ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))
5		ordtypelem.5	. . . . . . . . . . . 12 ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}
6		ordtypelem.6	. . . . . . . . . . . 12 ⊢ 𝑂 = OrdIso(𝑅, 𝐴)
7		ordtypelem.7	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 We 𝐴)
8		ordtypelem.8	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 Se 𝐴)
9	2, 3, 4, 5, 6, 7, 8	ordtypelem4 9518	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
10	9	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
11	10	fdmd 6720	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → dom 𝑂 = (𝑇 ∩ dom 𝐹))
12		inss1 4133	. . . . . . . . . 10 ⊢ (𝑇 ∩ dom 𝐹) ⊆ 𝑇
13	2, 3, 4, 5, 6, 7, 8	ordtypelem2 9516	. . . . . . . . . . . 12 ⊢ (𝜑 → Ord 𝑇)
14	13	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → Ord 𝑇)
15		ordsson 7797	. . . . . . . . . . 11 ⊢ (Ord 𝑇 → 𝑇 ⊆ On)
16	14, 15	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → 𝑇 ⊆ On)
17	12, 16	sstrid 3919	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑇 ∩ dom 𝐹) ⊆ On)
18	11, 17	eqsstrd 3942	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → dom 𝑂 ⊆ On)
19	18	sseld 3907	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑀 ∈ dom 𝑂 → 𝑀 ∈ On))
20		eleq1 2848	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑎 ∈ dom 𝑂 ↔ 𝑏 ∈ dom 𝑂))
21		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → (𝑂‘𝑎) = (𝑂‘𝑏))
22	21	breq1d 5112	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → ((𝑂‘𝑎)𝑅𝑁 ↔ (𝑂‘𝑏)𝑅𝑁))
23	20, 22	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ((𝑎 ∈ dom 𝑂 → (𝑂‘𝑎)𝑅𝑁) ↔ (𝑏 ∈ dom 𝑂 → (𝑂‘𝑏)𝑅𝑁)))
24	23	imbi2d 343	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (𝑂‘𝑎)𝑅𝑁)) ↔ ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑏 ∈ dom 𝑂 → (𝑂‘𝑏)𝑅𝑁))))
25		eleq1 2848	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑀 → (𝑎 ∈ dom 𝑂 ↔ 𝑀 ∈ dom 𝑂))
26		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑀 → (𝑂‘𝑎) = (𝑂‘𝑀))
27	26	breq1d 5112	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑀 → ((𝑂‘𝑎)𝑅𝑁 ↔ (𝑂‘𝑀)𝑅𝑁))
28	25, 27	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑎 = 𝑀 → ((𝑎 ∈ dom 𝑂 → (𝑂‘𝑎)𝑅𝑁) ↔ (𝑀 ∈ dom 𝑂 → (𝑂‘𝑀)𝑅𝑁)))
29	28	imbi2d 343	. . . . . . . . 9 ⊢ (𝑎 = 𝑀 → (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (𝑂‘𝑎)𝑅𝑁)) ↔ ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑀 ∈ dom 𝑂 → (𝑂‘𝑀)𝑅𝑁))))
30		r19.21v 3187	. . . . . . . . . 10 ⊢ (∀𝑏 ∈ 𝑎 ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑏 ∈ dom 𝑂 → (𝑂‘𝑏)𝑅𝑁)) ↔ ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → ∀𝑏 ∈ 𝑎 (𝑏 ∈ dom 𝑂 → (𝑂‘𝑏)𝑅𝑁)))
31	2	tfr1a 8405	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹)
32	31	simpri 491	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Lim dom 𝐹
33		limord 6424	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Lim dom 𝐹 → Ord dom 𝐹)
34	32, 33	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ord dom 𝐹
35		ordin 6393	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Ord 𝑇 ∧ Ord dom 𝐹) → Ord (𝑇 ∩ dom 𝐹))
36	14, 34, 35	sylancl 598	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → Ord (𝑇 ∩ dom 𝐹))
37		ordeq 6369	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (dom 𝑂 = (𝑇 ∩ dom 𝐹) → (Ord dom 𝑂 ↔ Ord (𝑇 ∩ dom 𝐹)))
38	11, 37	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (Ord dom 𝑂 ↔ Ord (𝑇 ∩ dom 𝐹)))
39	36, 38	mpbird 260	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → Ord dom 𝑂)
40		ordelss 6378	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Ord dom 𝑂 ∧ 𝑎 ∈ dom 𝑂) → 𝑎 ⊆ dom 𝑂)
41	39, 40	sylan 592	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ 𝑎 ∈ dom 𝑂) → 𝑎 ⊆ dom 𝑂)
42	41	sselda 3908	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ 𝑎 ∈ dom 𝑂) ∧ 𝑏 ∈ 𝑎) → 𝑏 ∈ dom 𝑂)
43		pm5.5 364	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ dom 𝑂 → ((𝑏 ∈ dom 𝑂 → (𝑂‘𝑏)𝑅𝑁) ↔ (𝑂‘𝑏)𝑅𝑁))
44	42, 43	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ 𝑎 ∈ dom 𝑂) ∧ 𝑏 ∈ 𝑎) → ((𝑏 ∈ dom 𝑂 → (𝑂‘𝑏)𝑅𝑁) ↔ (𝑂‘𝑏)𝑅𝑁))
45	44	ralbidva 3183	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ 𝑎 ∈ dom 𝑂) → (∀𝑏 ∈ 𝑎 (𝑏 ∈ dom 𝑂 → (𝑂‘𝑏)𝑅𝑁) ↔ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁))
46		eldifn 4068	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ (𝐴 ∖ ran 𝑂) → ¬ 𝑁 ∈ ran 𝑂)
47	46	ad2antlr 740	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → ¬ 𝑁 ∈ ran 𝑂)
48	9	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
49	48	ffnd 6710	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → 𝑂 Fn (𝑇 ∩ dom 𝐹))
50		simprl 783	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → 𝑎 ∈ dom 𝑂)
51	48	fdmd 6720	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → dom 𝑂 = (𝑇 ∩ dom 𝐹))
52	50, 51	eleqtrd 2862	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → 𝑎 ∈ (𝑇 ∩ dom 𝐹))
53		fnfvelrn 7080	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑂 Fn (𝑇 ∩ dom 𝐹) ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝑂‘𝑎) ∈ ran 𝑂)
54	49, 52, 53	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → (𝑂‘𝑎) ∈ ran 𝑂)
55		eleq1 2848	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑂‘𝑎) = 𝑁 → ((𝑂‘𝑎) ∈ ran 𝑂 ↔ 𝑁 ∈ ran 𝑂))
56	54, 55	syl5ibcom 248	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → ((𝑂‘𝑎) = 𝑁 → 𝑁 ∈ ran 𝑂))
57	47, 56	mtod 201	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → ¬ (𝑂‘𝑎) = 𝑁)
58		breq1 5105	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑁 → (𝑢𝑅(𝑂‘𝑎) ↔ 𝑁𝑅(𝑂‘𝑎)))
59	58	notbid 321	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑁 → (¬ 𝑢𝑅(𝑂‘𝑎) ↔ ¬ 𝑁𝑅(𝑂‘𝑎)))
60	2, 3, 4, 5, 6, 7, 8	ordtypelem1 9515	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑂 = (𝐹 ↾ 𝑇))
61	60	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → 𝑂 = (𝐹 ↾ 𝑇))
62	61	fveq1d 6887	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → (𝑂‘𝑎) = ((𝐹 ↾ 𝑇)‘𝑎))
63	52	elin1d 4101	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → 𝑎 ∈ 𝑇)
64	63	fvresd 6905	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → ((𝐹 ↾ 𝑇)‘𝑎) = (𝐹‘𝑎))
65	62, 64	eqtrd 2795	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → (𝑂‘𝑎) = (𝐹‘𝑎))
66		simpll 779	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → 𝜑)
67	2, 3, 4, 5, 6, 7, 8	ordtypelem3 9517	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹‘𝑎) ∈ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
68	66, 52, 67	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → (𝐹‘𝑎) ∈ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
69	65, 68	eqeltrd 2860	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → (𝑂‘𝑎) ∈ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
70		breq2 5106	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = (𝑂‘𝑎) → (𝑢𝑅𝑣 ↔ 𝑢𝑅(𝑂‘𝑎)))
71	70	notbid 321	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = (𝑂‘𝑎) → (¬ 𝑢𝑅𝑣 ↔ ¬ 𝑢𝑅(𝑂‘𝑎)))
72	71	ralbidv 3185	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = (𝑂‘𝑎) → (∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣 ↔ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅(𝑂‘𝑎)))
73	72	elrab 3644	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑂‘𝑎) ∈ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} ↔ ((𝑂‘𝑎) ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ∧ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅(𝑂‘𝑎)))
74	73	simprbi 503	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑂‘𝑎) ∈ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} → ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅(𝑂‘𝑎))
75	69, 74	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅(𝑂‘𝑎))
76		breq2 5106	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑁 → (𝑗𝑅𝑤 ↔ 𝑗𝑅𝑁))
77	76	ralbidv 3185	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑁 → (∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤 ↔ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑁))
78		eldifi 4067	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (𝐴 ∖ ran 𝑂) → 𝑁 ∈ 𝐴)
79	78	ad2antlr 740	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → 𝑁 ∈ 𝐴)
80		simprr 785	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)
81	41	adantrr 730	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → 𝑎 ⊆ dom 𝑂)
82	48, 81	fssdmd 6728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → 𝑎 ⊆ (𝑇 ∩ dom 𝐹))
83	82, 12	sstrdi 3920	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → 𝑎 ⊆ 𝑇)
84		fveq1 6884	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑂 = (𝐹 ↾ 𝑇) → (𝑂‘𝑏) = ((𝐹 ↾ 𝑇)‘𝑏))
85		ssel2 3903	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑎 ⊆ 𝑇 ∧ 𝑏 ∈ 𝑎) → 𝑏 ∈ 𝑇)
86	85	fvresd 6905	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎 ⊆ 𝑇 ∧ 𝑏 ∈ 𝑎) → ((𝐹 ↾ 𝑇)‘𝑏) = (𝐹‘𝑏))
87	84, 86	sylan9eq 2815	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑂 = (𝐹 ↾ 𝑇) ∧ (𝑎 ⊆ 𝑇 ∧ 𝑏 ∈ 𝑎)) → (𝑂‘𝑏) = (𝐹‘𝑏))
88	87	anassrs 473	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑂 = (𝐹 ↾ 𝑇) ∧ 𝑎 ⊆ 𝑇) ∧ 𝑏 ∈ 𝑎) → (𝑂‘𝑏) = (𝐹‘𝑏))
89	88	breq1d 5112	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑂 = (𝐹 ↾ 𝑇) ∧ 𝑎 ⊆ 𝑇) ∧ 𝑏 ∈ 𝑎) → ((𝑂‘𝑏)𝑅𝑁 ↔ (𝐹‘𝑏)𝑅𝑁))
90	89	ralbidva 3183	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑂 = (𝐹 ↾ 𝑇) ∧ 𝑎 ⊆ 𝑇) → (∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁 ↔ ∀𝑏 ∈ 𝑎 (𝐹‘𝑏)𝑅𝑁))
91	61, 83, 90	syl2anc 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → (∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁 ↔ ∀𝑏 ∈ 𝑎 (𝐹‘𝑏)𝑅𝑁))
92	80, 91	mpbid 235	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → ∀𝑏 ∈ 𝑎 (𝐹‘𝑏)𝑅𝑁)
93	31	simpli 489	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Fun 𝐹
94		funfn 6570	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
95	93, 94	mpbi 233	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐹 Fn dom 𝐹
96		inss2 4134	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑇 ∩ dom 𝐹) ⊆ dom 𝐹
97	82, 96	sstrdi 3920	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → 𝑎 ⊆ dom 𝐹)
98		breq1 5105	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = (𝐹‘𝑏) → (𝑗𝑅𝑁 ↔ (𝐹‘𝑏)𝑅𝑁))
99	98	ralima 7243	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝑎 ⊆ dom 𝐹) → (∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑁 ↔ ∀𝑏 ∈ 𝑎 (𝐹‘𝑏)𝑅𝑁))
100	95, 97, 99	sylancr 599	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → (∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑁 ↔ ∀𝑏 ∈ 𝑎 (𝐹‘𝑏)𝑅𝑁))
101	92, 100	mpbird 260	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑁)
102	77, 79, 101	elrabd 3646	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → 𝑁 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤})
103	59, 75, 102	rspcdva 3577	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → ¬ 𝑁𝑅(𝑂‘𝑎))
104		weso 5641	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
105	7, 104	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑅 Or 𝐴)
106	105	ad2antrr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → 𝑅 Or 𝐴)
107	48, 52	ffvelcdmd 7085	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → (𝑂‘𝑎) ∈ 𝐴)
108		sotric 5588	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 Or 𝐴 ∧ ((𝑂‘𝑎) ∈ 𝐴 ∧ 𝑁 ∈ 𝐴)) → ((𝑂‘𝑎)𝑅𝑁 ↔ ¬ ((𝑂‘𝑎) = 𝑁 ∨ 𝑁𝑅(𝑂‘𝑎))))
109	106, 107, 79, 108	syl12anc 850	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → ((𝑂‘𝑎)𝑅𝑁 ↔ ¬ ((𝑂‘𝑎) = 𝑁 ∨ 𝑁𝑅(𝑂‘𝑎))))
110		ioran 999	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ ((𝑂‘𝑎) = 𝑁 ∨ 𝑁𝑅(𝑂‘𝑎)) ↔ (¬ (𝑂‘𝑎) = 𝑁 ∧ ¬ 𝑁𝑅(𝑂‘𝑎)))
111	109, 110	bitrdi 290	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → ((𝑂‘𝑎)𝑅𝑁 ↔ (¬ (𝑂‘𝑎) = 𝑁 ∧ ¬ 𝑁𝑅(𝑂‘𝑎))))
112	57, 103, 111	mpbir2and 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ (𝑎 ∈ dom 𝑂 ∧ ∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁)) → (𝑂‘𝑎)𝑅𝑁)
113	112	expr 462	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ 𝑎 ∈ dom 𝑂) → (∀𝑏 ∈ 𝑎 (𝑂‘𝑏)𝑅𝑁 → (𝑂‘𝑎)𝑅𝑁))
114	45, 113	sylbid 243	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) ∧ 𝑎 ∈ dom 𝑂) → (∀𝑏 ∈ 𝑎 (𝑏 ∈ dom 𝑂 → (𝑂‘𝑏)𝑅𝑁) → (𝑂‘𝑎)𝑅𝑁))
115	114	ex 418	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (∀𝑏 ∈ 𝑎 (𝑏 ∈ dom 𝑂 → (𝑂‘𝑏)𝑅𝑁) → (𝑂‘𝑎)𝑅𝑁)))
116	115	com23 87	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (∀𝑏 ∈ 𝑎 (𝑏 ∈ dom 𝑂 → (𝑂‘𝑏)𝑅𝑁) → (𝑎 ∈ dom 𝑂 → (𝑂‘𝑎)𝑅𝑁)))
117	116	a2i 15	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → ∀𝑏 ∈ 𝑎 (𝑏 ∈ dom 𝑂 → (𝑂‘𝑏)𝑅𝑁)) → ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (𝑂‘𝑎)𝑅𝑁)))
118	117	a1i 11	. . . . . . . . . 10 ⊢ (𝑎 ∈ On → (((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → ∀𝑏 ∈ 𝑎 (𝑏 ∈ dom 𝑂 → (𝑂‘𝑏)𝑅𝑁)) → ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (𝑂‘𝑎)𝑅𝑁))))
119	30, 118	biimtrid 245	. . . . . . . . 9 ⊢ (𝑎 ∈ On → (∀𝑏 ∈ 𝑎 ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑏 ∈ dom 𝑂 → (𝑂‘𝑏)𝑅𝑁)) → ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑎 ∈ dom 𝑂 → (𝑂‘𝑎)𝑅𝑁))))
120	24, 29, 119	tfis3 7869	. . . . . . . 8 ⊢ (𝑀 ∈ On → ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑀 ∈ dom 𝑂 → (𝑂‘𝑀)𝑅𝑁)))
121	120	com3l 90	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑀 ∈ dom 𝑂 → (𝑀 ∈ On → (𝑂‘𝑀)𝑅𝑁)))
122	19, 121	mpdd 44	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (𝐴 ∖ ran 𝑂)) → (𝑀 ∈ dom 𝑂 → (𝑂‘𝑀)𝑅𝑁))
123	1, 122	sylan2br 607	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ∈ ran 𝑂)) → (𝑀 ∈ dom 𝑂 → (𝑂‘𝑀)𝑅𝑁))
124	123	anassrs 473	. . . 4 ⊢ (((𝜑 ∧ 𝑁 ∈ 𝐴) ∧ ¬ 𝑁 ∈ ran 𝑂) → (𝑀 ∈ dom 𝑂 → (𝑂‘𝑀)𝑅𝑁))
125	124	impancom 457	. . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ 𝐴) ∧ 𝑀 ∈ dom 𝑂) → (¬ 𝑁 ∈ ran 𝑂 → (𝑂‘𝑀)𝑅𝑁))
126	125	orrd 877	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ 𝐴) ∧ 𝑀 ∈ dom 𝑂) → (𝑁 ∈ ran 𝑂 ∨ (𝑂‘𝑀)𝑅𝑁))
127	126	orcomd 885	1 ⊢ (((𝜑 ∧ 𝑁 ∈ 𝐴) ∧ 𝑀 ∈ dom 𝑂) → ((𝑂‘𝑀)𝑅𝑁 ∨ 𝑁 ∈ ran 𝑂))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3895   ∖ cdif 4045   ∩ cin 4089   class class class wbr 5102   ↦ cmpt 5185   Or wor 5557   Se wse 5601   We wwe 5602  dom cdm 5650  ran crn 5651   ↾ cres 5652   “ cima 5653  Ord word 6361  Oncon0 6362  Lim wlim 6363  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  ℩crio 7376  recscrecs 8381  OrdIsocoi 9506

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-oi 9507

This theorem is used by:  ordtypelem9  9523  ordtypelem10  9524  oiiniseg  9530