Theorem ordtypelem3 8425

Description: Lemma for ordtype 8437. (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		inss2 3834	. . . . 5 ⊢ (𝑇 ∩ dom 𝐹) ⊆ dom 𝐹
2		simpr 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → 𝑀 ∈ (𝑇 ∩ dom 𝐹))
3	1, 2	sseldi 3601	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → 𝑀 ∈ dom 𝐹)
4		ordtypelem.1	. . . . 5 ⊢ 𝐹 = recs(𝐺)
5	4	tfr2a 7491	. . . 4 ⊢ (𝑀 ∈ dom 𝐹 → (𝐹‘𝑀) = (𝐺‘(𝐹 ↾ 𝑀)))
6	3, 5	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹‘𝑀) = (𝐺‘(𝐹 ↾ 𝑀)))
7	4	tfr1a 7490	. . . . . . . . 9 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹)
8	7	simpri 478	. . . . . . . 8 ⊢ Lim dom 𝐹
9		limord 5784	. . . . . . . 8 ⊢ (Lim dom 𝐹 → Ord dom 𝐹)
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ Ord dom 𝐹
11		ordelord 5745	. . . . . . 7 ⊢ ((Ord dom 𝐹 ∧ 𝑀 ∈ dom 𝐹) → Ord 𝑀)
12	10, 3, 11	sylancr 695	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → Ord 𝑀)
13	4	tfr2b 7492	. . . . . 6 ⊢ (Ord 𝑀 → (𝑀 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝑀) ∈ V))
14	12, 13	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝑀 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝑀) ∈ V))
15	3, 14	mpbid 222	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹 ↾ 𝑀) ∈ V)
16		ordtypelem.2	. . . . . . 7 ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
17		rneq 5351	. . . . . . . . . 10 ⊢ (ℎ = (𝐹 ↾ 𝑀) → ran ℎ = ran (𝐹 ↾ 𝑀))
18		df-ima 5127	. . . . . . . . . 10 ⊢ (𝐹 “ 𝑀) = ran (𝐹 ↾ 𝑀)
19	17, 18	syl6eqr 2674	. . . . . . . . 9 ⊢ (ℎ = (𝐹 ↾ 𝑀) → ran ℎ = (𝐹 “ 𝑀))
20	19	raleqdv 3144	. . . . . . . 8 ⊢ (ℎ = (𝐹 ↾ 𝑀) → (∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤 ↔ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤))
21	20	rabbidv 3189	. . . . . . 7 ⊢ (ℎ = (𝐹 ↾ 𝑀) → {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤})
22	16, 21	syl5eq 2668	. . . . . 6 ⊢ (ℎ = (𝐹 ↾ 𝑀) → 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤})
23	22	raleqdv 3144	. . . . . 6 ⊢ (ℎ = (𝐹 ↾ 𝑀) → (∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣 ↔ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
24	22, 23	riotaeqbidv 6614	. . . . 5 ⊢ (ℎ = (𝐹 ↾ 𝑀) → (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣) = (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
25		ordtypelem.3	. . . . 5 ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))
26		riotaex 6615	. . . . 5 ⊢ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣) ∈ V
27	24, 25, 26	fvmpt 6282	. . . 4 ⊢ ((𝐹 ↾ 𝑀) ∈ V → (𝐺‘(𝐹 ↾ 𝑀)) = (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
28	15, 27	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))
29	6, 28	eqtrd 2656	. 2 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹‘𝑀) = (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
30		ordtypelem.7	. . . . 5 ⊢ (𝜑 → 𝑅 We 𝐴)
31	30	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → 𝑅 We 𝐴)
32		ordtypelem.8	. . . . 5 ⊢ (𝜑 → 𝑅 Se 𝐴)
33	32	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → 𝑅 Se 𝐴)
34		ssrab2 3687	. . . . 5 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ⊆ 𝐴
35	34	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ⊆ 𝐴)
36		inss1 3833	. . . . . . . 8 ⊢ (𝑇 ∩ dom 𝐹) ⊆ 𝑇
37	36, 2	sseldi 3601	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → 𝑀 ∈ 𝑇)
38		imaeq2 5462	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑀 → (𝐹 “ 𝑥) = (𝐹 “ 𝑀))
39	38	raleqdv 3144	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡 ↔ ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡))
40	39	rexbidv 3052	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡 ↔ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡))
41		ordtypelem.5	. . . . . . . . 9 ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}
42	40, 41	elrab2 3366	. . . . . . . 8 ⊢ (𝑀 ∈ 𝑇 ↔ (𝑀 ∈ On ∧ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡))
43	42	simprbi 480	. . . . . . 7 ⊢ (𝑀 ∈ 𝑇 → ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡)
44	37, 43	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡)
45		breq1 4656	. . . . . . . . 9 ⊢ (𝑗 = 𝑧 → (𝑗𝑅𝑤 ↔ 𝑧𝑅𝑤))
46	45	cbvralv 3171	. . . . . . . 8 ⊢ (∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤 ↔ ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑤)
47		breq2 4657	. . . . . . . . 9 ⊢ (𝑤 = 𝑡 → (𝑧𝑅𝑤 ↔ 𝑧𝑅𝑡))
48	47	ralbidv 2986	. . . . . . . 8 ⊢ (𝑤 = 𝑡 → (∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑤 ↔ ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡))
49	46, 48	syl5bb 272	. . . . . . 7 ⊢ (𝑤 = 𝑡 → (∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤 ↔ ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡))
50	49	cbvrexv 3172	. . . . . 6 ⊢ (∃𝑤 ∈ 𝐴 ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤 ↔ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡)
51	44, 50	sylibr 224	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → ∃𝑤 ∈ 𝐴 ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤)
52		rabn0 3958	. . . . 5 ⊢ ({𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ≠ ∅ ↔ ∃𝑤 ∈ 𝐴 ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤)
53	51, 52	sylibr 224	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ≠ ∅)
54		wereu2 5111	. . . 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)
55	31, 33, 35, 53, 54	syl22anc 1327	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → ∃!𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)
56		riotacl2 6624	. . 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})
57	55, 56	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})
58	29, 57	eqeltrd 2701	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  ∃!wreu 2914  {crab 2916  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915   class class class wbr 4653   ↦ cmpt 4729   Se wse 5071   We wwe 5072  dom cdm 5114  ran crn 5115   ↾ cres 5116   “ cima 5117  Ord word 5722  Oncon0 5723  Lim wlim 5724  Fun wfun 5882  ‘cfv 5888  ℩crio 6610  recscrecs 7467  OrdIsocoi 8414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-wrecs 7407  df-recs 7468

This theorem is referenced by:  ordtypelem4  8426  ordtypelem6  8428  ordtypelem7  8429