Theorem ordtypelem3 9534

Description: Lemma for ordtype 9546. (Contributed by Mario Carneiro, 24-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
ordtypelem3	⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹‘𝑀) ∈ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})

Distinct variable groups:   𝑣,𝑢,𝐶   ℎ,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧,𝑀   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)   𝑂(𝑧,𝑤,ℎ,𝑗)

Proof of Theorem ordtypelem3

Step	Hyp	Ref	Expression
1		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → 𝑀 ∈ (𝑇 ∩ dom 𝐹))
2	1	elin2d 4180	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → 𝑀 ∈ dom 𝐹)
3		ordtypelem.1	. . . . 5 ⊢ 𝐹 = recs(𝐺)
4	3	tfr2a 8409	. . . 4 ⊢ (𝑀 ∈ dom 𝐹 → (𝐹‘𝑀) = (𝐺‘(𝐹 ↾ 𝑀)))
5	2, 4	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹‘𝑀) = (𝐺‘(𝐹 ↾ 𝑀)))
6	3	tfr1a 8408	. . . . . . . . 9 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹)
7	6	simpri 485	. . . . . . . 8 ⊢ Lim dom 𝐹
8		limord 6413	. . . . . . . 8 ⊢ (Lim dom 𝐹 → Ord dom 𝐹)
9	7, 8	ax-mp 5	. . . . . . 7 ⊢ Ord dom 𝐹
10		ordelord 6374	. . . . . . 7 ⊢ ((Ord dom 𝐹 ∧ 𝑀 ∈ dom 𝐹) → Ord 𝑀)
11	9, 2, 10	sylancr 587	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → Ord 𝑀)
12	3	tfr2b 8410	. . . . . 6 ⊢ (Ord 𝑀 → (𝑀 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝑀) ∈ V))
13	11, 12	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝑀 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝑀) ∈ V))
14	2, 13	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹 ↾ 𝑀) ∈ V)
15		ordtypelem.2	. . . . . . 7 ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
16		rneq 5916	. . . . . . . . . 10 ⊢ (ℎ = (𝐹 ↾ 𝑀) → ran ℎ = ran (𝐹 ↾ 𝑀))
17		df-ima 5667	. . . . . . . . . 10 ⊢ (𝐹 “ 𝑀) = ran (𝐹 ↾ 𝑀)
18	16, 17	eqtr4di 2788	. . . . . . . . 9 ⊢ (ℎ = (𝐹 ↾ 𝑀) → ran ℎ = (𝐹 “ 𝑀))
19	18	raleqdv 3305	. . . . . . . 8 ⊢ (ℎ = (𝐹 ↾ 𝑀) → (∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤 ↔ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤))
20	19	rabbidv 3423	. . . . . . 7 ⊢ (ℎ = (𝐹 ↾ 𝑀) → {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤})
21	15, 20	eqtrid 2782	. . . . . 6 ⊢ (ℎ = (𝐹 ↾ 𝑀) → 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤})
22	21	raleqdv 3305	. . . . . 6 ⊢ (ℎ = (𝐹 ↾ 𝑀) → (∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣 ↔ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
23	21, 22	riotaeqbidv 7365	. . . . 5 ⊢ (ℎ = (𝐹 ↾ 𝑀) → (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣) = (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
24		ordtypelem.3	. . . . 5 ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))
25		riotaex 7366	. . . . 5 ⊢ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣) ∈ V
26	23, 24, 25	fvmpt 6986	. . . 4 ⊢ ((𝐹 ↾ 𝑀) ∈ V → (𝐺‘(𝐹 ↾ 𝑀)) = (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
27	14, 26	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐺‘(𝐹 ↾ 𝑀)) = (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
28	5, 27	eqtrd 2770	. 2 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹‘𝑀) = (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
29		ordtypelem.7	. . . . 5 ⊢ (𝜑 → 𝑅 We 𝐴)
30	29	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → 𝑅 We 𝐴)
31		ordtypelem.8	. . . . 5 ⊢ (𝜑 → 𝑅 Se 𝐴)
32	31	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → 𝑅 Se 𝐴)
33		ssrab2 4055	. . . . 5 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ⊆ 𝐴
34	33	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ⊆ 𝐴)
35	1	elin1d 4179	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → 𝑀 ∈ 𝑇)
36		imaeq2 6043	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑀 → (𝐹 “ 𝑥) = (𝐹 “ 𝑀))
37	36	raleqdv 3305	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡 ↔ ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡))
38	37	rexbidv 3164	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡 ↔ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡))
39		ordtypelem.5	. . . . . . . . 9 ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}
40	38, 39	elrab2 3674	. . . . . . . 8 ⊢ (𝑀 ∈ 𝑇 ↔ (𝑀 ∈ On ∧ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡))
41	40	simprbi 496	. . . . . . 7 ⊢ (𝑀 ∈ 𝑇 → ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡)
42	35, 41	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡)
43		breq1 5122	. . . . . . . . 9 ⊢ (𝑗 = 𝑧 → (𝑗𝑅𝑤 ↔ 𝑧𝑅𝑤))
44	43	cbvralvw 3220	. . . . . . . 8 ⊢ (∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤 ↔ ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑤)
45		breq2 5123	. . . . . . . . 9 ⊢ (𝑤 = 𝑡 → (𝑧𝑅𝑤 ↔ 𝑧𝑅𝑡))
46	45	ralbidv 3163	. . . . . . . 8 ⊢ (𝑤 = 𝑡 → (∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑤 ↔ ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡))
47	44, 46	bitrid 283	. . . . . . 7 ⊢ (𝑤 = 𝑡 → (∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤 ↔ ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡))
48	47	cbvrexvw 3221	. . . . . 6 ⊢ (∃𝑤 ∈ 𝐴 ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤 ↔ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡)
49	42, 48	sylibr 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → ∃𝑤 ∈ 𝐴 ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤)
50		rabn0 4364	. . . . 5 ⊢ ({𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ≠ ∅ ↔ ∃𝑤 ∈ 𝐴 ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤)
51	49, 50	sylibr 234	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ≠ ∅)
52		wereu2 5651	. . . 4 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ ({𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ⊆ 𝐴 ∧ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ≠ ∅)) → ∃!𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)
53	30, 32, 34, 51, 52	syl22anc 838	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → ∃!𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)
54		riotacl2 7378	. . 3 ⊢ (∃!𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣 → (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣) ∈ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
55	53, 54	syl 17	. 2 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣) ∈ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
56	28, 55	eqeltrd 2834	1 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹‘𝑀) ∈ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  ∃!wreu 3357  {crab 3415  Vcvv 3459   ∩ cin 3925   ⊆ wss 3926  ∅c0 4308   class class class wbr 5119   ↦ cmpt 5201   Se wse 5604   We wwe 5605  dom cdm 5654  ran crn 5655   ↾ cres 5656   “ cima 5657  Ord word 6351  Oncon0 6352  Lim wlim 6353  Fun wfun 6525  ‘cfv 6531  ℩crio 7361  recscrecs 8384  OrdIsocoi 9523

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385

This theorem is referenced by:  ordtypelem4  9535  ordtypelem6  9537  ordtypelem7  9538