Theorem ordtypelem3 9498

Description: Lemma for ordtype 9510. (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 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → 𝑀 ∈ (𝑇 ∩ dom 𝐹))
2	1	elin2d 4151	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → 𝑀 ∈ dom 𝐹)
3		ordtypelem.1	. . . . 5 ⊢ 𝐹 = recs(𝐺)
4	3	tfr2a 8387	. . . 4 ⊢ (𝑀 ∈ dom 𝐹 → (𝐹‘𝑀) = (𝐺‘(𝐹 ↾ 𝑀)))
5	2, 4	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹‘𝑀) = (𝐺‘(𝐹 ↾ 𝑀)))
6	3	tfr1a 8386	. . . . . . . . 9 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹)
7	6	simpri 491	. . . . . . . 8 ⊢ Lim dom 𝐹
8		limord 6417	. . . . . . . 8 ⊢ (Lim dom 𝐹 → Ord dom 𝐹)
9	7, 8	ax-mp 5	. . . . . . 7 ⊢ Ord dom 𝐹
10		ordelord 6377	. . . . . . 7 ⊢ ((Ord dom 𝐹 ∧ 𝑀 ∈ dom 𝐹) → Ord 𝑀)
11	9, 2, 10	sylancr 599	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → Ord 𝑀)
12	3	tfr2b 8388	. . . . . 6 ⊢ (Ord 𝑀 → (𝑀 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝑀) ∈ V))
13	11, 12	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝑀 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝑀) ∈ V))
14	2, 13	mpbid 235	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹 ↾ 𝑀) ∈ V)
15		ordtypelem.2	. . . . . . 7 ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
16		rneq 5918	. . . . . . . . . 10 ⊢ (ℎ = (𝐹 ↾ 𝑀) → ran ℎ = ran (𝐹 ↾ 𝑀))
17		df-ima 5664	. . . . . . . . . 10 ⊢ (𝐹 “ 𝑀) = ran (𝐹 ↾ 𝑀)
18	16, 17	eqtr4di 2814	. . . . . . . . 9 ⊢ (ℎ = (𝐹 ↾ 𝑀) → ran ℎ = (𝐹 “ 𝑀))
19	18	raleqdv 3320	. . . . . . . 8 ⊢ (ℎ = (𝐹 ↾ 𝑀) → (∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤 ↔ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤))
20	19	rabbidv 3420	. . . . . . 7 ⊢ (ℎ = (𝐹 ↾ 𝑀) → {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤})
21	15, 20	eqtrid 2808	. . . . . 6 ⊢ (ℎ = (𝐹 ↾ 𝑀) → 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤})
22	21	raleqdv 3320	. . . . . 6 ⊢ (ℎ = (𝐹 ↾ 𝑀) → (∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣 ↔ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
23	21, 22	riotaeqbidv 7372	. . . . 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 7373	. . . . 5 ⊢ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣) ∈ V
26	23, 24, 25	fvmpt 6985	. . . 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 18	. . 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 2796	. 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 486	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → 𝑅 We 𝐴)
31		ordtypelem.8	. . . . 5 ⊢ (𝜑 → 𝑅 Se 𝐴)
32	31	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → 𝑅 Se 𝐴)
33		ssrab2 4028	. . . . 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 4150	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → 𝑀 ∈ 𝑇)
36		imaeq2 6050	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑀 → (𝐹 “ 𝑥) = (𝐹 “ 𝑀))
37	36	raleqdv 3320	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡 ↔ ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡))
38	37	rexbidv 3187	. . . . . . . . 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 3649	. . . . . . . 8 ⊢ (𝑀 ∈ 𝑇 ↔ (𝑀 ∈ On ∧ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡))
41	40	simprbi 503	. . . . . . 7 ⊢ (𝑀 ∈ 𝑇 → ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡)
42	35, 41	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡)
43		breq1 5106	. . . . . . . . 9 ⊢ (𝑗 = 𝑧 → (𝑗𝑅𝑤 ↔ 𝑧𝑅𝑤))
44	43	cbvralvw 3241	. . . . . . . 8 ⊢ (∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤 ↔ ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑤)
45		breq2 5107	. . . . . . . . 9 ⊢ (𝑤 = 𝑡 → (𝑧𝑅𝑤 ↔ 𝑧𝑅𝑡))
46	45	ralbidv 3186	. . . . . . . 8 ⊢ (𝑤 = 𝑡 → (∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑤 ↔ ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡))
47	44, 46	bitrid 286	. . . . . . 7 ⊢ (𝑤 = 𝑡 → (∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤 ↔ ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡))
48	47	cbvrexvw 3242	. . . . . 6 ⊢ (∃𝑤 ∈ 𝐴 ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤 ↔ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑀)𝑧𝑅𝑡)
49	42, 48	sylibr 237	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → ∃𝑤 ∈ 𝐴 ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤)
50		rabn0 4339	. . . . 5 ⊢ ({𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ≠ ∅ ↔ ∃𝑤 ∈ 𝐴 ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤)
51	49, 50	sylibr 237	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ≠ ∅)
52		wereu2 5648	. . . 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 852	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → ∃!𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)
54		riotacl2 7385	. . 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 18	. 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 2861	1 ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹‘𝑀) ∈ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  ∃!wreu 3364  {crab 3413  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279   class class class wbr 5103   ↦ cmpt 5186   Se wse 5602   We wwe 5603  dom cdm 5651  ran crn 5652   ↾ cres 5653   “ cima 5654  Ord word 6354  Oncon0 6355  Lim wlim 6356  Fun wfun 6525  ‘cfv 6531  ℩crio 7368  recscrecs 8362  OrdIsocoi 9487

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363

This theorem is used by:  ordtypelem4  9499  ordtypelem6  9501  ordtypelem7  9502