Theorem ordtypelem9 9498

Description: Lemma for ordtype 9504. Either the function OrdIso is an isomorphism onto all of 𝐴, or OrdIso is not a set, which by oif 9502 implies that either ran 𝑂 ⊆ 𝐴 is a proper class or dom 𝑂 = On. (Contributed by Mario Carneiro, 25-Jun-2015.) (Revised by AV, 28-Jul-2024.)

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)
ordtypelem9.1	⊢ (𝜑 → 𝑂 ∈ 𝑉)

Assertion

Ref	Expression
ordtypelem9	⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))

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

Proof of Theorem ordtypelem9

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

Step	Hyp	Ref	Expression
1		ordtypelem.1	. . 3 ⊢ 𝐹 = recs(𝐺)
2		ordtypelem.2	. . 3 ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
3		ordtypelem.3	. . 3 ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))
4		ordtypelem.5	. . 3 ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}
5		ordtypelem.6	. . 3 ⊢ 𝑂 = OrdIso(𝑅, 𝐴)
6		ordtypelem.7	. . 3 ⊢ (𝜑 → 𝑅 We 𝐴)
7		ordtypelem.8	. . 3 ⊢ (𝜑 → 𝑅 Se 𝐴)
8	1, 2, 3, 4, 5, 6, 7	ordtypelem8 9497	. 2 ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
9	1, 2, 3, 4, 5, 6, 7	ordtypelem4 9493	. . . . 5 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
10	9	frnd 6711	. . . 4 ⊢ (𝜑 → ran 𝑂 ⊆ 𝐴)
11	1, 2, 3, 4, 5, 6, 7	ordtypelem2 9491	. . . . . . . . . . 11 ⊢ (𝜑 → Ord 𝑇)
12		ordirr 6375	. . . . . . . . . . 11 ⊢ (Ord 𝑇 → ¬ 𝑇 ∈ 𝑇)
13	11, 12	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑇 ∈ 𝑇)
14	1	tfr1a 8383	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹)
15	14	simpri 491	. . . . . . . . . . . . 13 ⊢ Lim dom 𝐹
16		limord 6419	. . . . . . . . . . . . 13 ⊢ (Lim dom 𝐹 → Ord dom 𝐹)
17	15, 16	ax-mp 5	. . . . . . . . . . . 12 ⊢ Ord dom 𝐹
18	1, 2, 3, 4, 5, 6, 7	ordtypelem1 9490	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑂 = (𝐹 ↾ 𝑇))
19		ordtypelem9.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑂 ∈ 𝑉)
20	19	elexd 3473	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑂 ∈ V)
21	18, 20	eqeltrrd 2861	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ↾ 𝑇) ∈ V)
22	1	tfr2b 8385	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑇 → (𝑇 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝑇) ∈ V))
23	11, 22	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑇 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝑇) ∈ V))
24	21, 23	mpbird 260	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ dom 𝐹)
25		ordelon 6381	. . . . . . . . . . . 12 ⊢ ((Ord dom 𝐹 ∧ 𝑇 ∈ dom 𝐹) → 𝑇 ∈ On)
26	17, 24, 25	sylancr 599	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ On)
27		imaeq2 6052	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑇 → (𝐹 “ 𝑎) = (𝐹 “ 𝑇))
28	27	raleqdv 3319	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑇 → (∀𝑐 ∈ (𝐹 “ 𝑎)𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹 “ 𝑇)𝑐𝑅𝑏))
29	28	rexbidv 3186	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑇 → (∃𝑏 ∈ 𝐴 ∀𝑐 ∈ (𝐹 “ 𝑎)𝑐𝑅𝑏 ↔ ∃𝑏 ∈ 𝐴 ∀𝑐 ∈ (𝐹 “ 𝑇)𝑐𝑅𝑏))
30		breq1 5106	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑐 → (𝑧𝑅𝑡 ↔ 𝑐𝑅𝑡))
31	30	cbvralvw 3240	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹 “ 𝑥)𝑐𝑅𝑡)
32		breq2 5107	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑏 → (𝑐𝑅𝑡 ↔ 𝑐𝑅𝑏))
33	32	ralbidv 3185	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑏 → (∀𝑐 ∈ (𝐹 “ 𝑥)𝑐𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹 “ 𝑥)𝑐𝑅𝑏))
34	31, 33	bitrid 286	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑏 → (∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹 “ 𝑥)𝑐𝑅𝑏))
35	34	cbvrexvw 3241	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡 ↔ ∃𝑏 ∈ 𝐴 ∀𝑐 ∈ (𝐹 “ 𝑥)𝑐𝑅𝑏)
36		imaeq2 6052	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → (𝐹 “ 𝑥) = (𝐹 “ 𝑎))
37	36	raleqdv 3319	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (∀𝑐 ∈ (𝐹 “ 𝑥)𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹 “ 𝑎)𝑐𝑅𝑏))
38	37	rexbidv 3186	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → (∃𝑏 ∈ 𝐴 ∀𝑐 ∈ (𝐹 “ 𝑥)𝑐𝑅𝑏 ↔ ∃𝑏 ∈ 𝐴 ∀𝑐 ∈ (𝐹 “ 𝑎)𝑐𝑅𝑏))
39	35, 38	bitrid 286	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → (∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡 ↔ ∃𝑏 ∈ 𝐴 ∀𝑐 ∈ (𝐹 “ 𝑎)𝑐𝑅𝑏))
40	39	cbvrabv 3422	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} = {𝑎 ∈ On ∣ ∃𝑏 ∈ 𝐴 ∀𝑐 ∈ (𝐹 “ 𝑎)𝑐𝑅𝑏}
41	4, 40	eqtri 2783	. . . . . . . . . . . . 13 ⊢ 𝑇 = {𝑎 ∈ On ∣ ∃𝑏 ∈ 𝐴 ∀𝑐 ∈ (𝐹 “ 𝑎)𝑐𝑅𝑏}
42	29, 41	elrab2 3649	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ 𝑇 ↔ (𝑇 ∈ On ∧ ∃𝑏 ∈ 𝐴 ∀𝑐 ∈ (𝐹 “ 𝑇)𝑐𝑅𝑏))
43	42	baib 545	. . . . . . . . . . 11 ⊢ (𝑇 ∈ On → (𝑇 ∈ 𝑇 ↔ ∃𝑏 ∈ 𝐴 ∀𝑐 ∈ (𝐹 “ 𝑇)𝑐𝑅𝑏))
44	26, 43	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 ∈ 𝑇 ↔ ∃𝑏 ∈ 𝐴 ∀𝑐 ∈ (𝐹 “ 𝑇)𝑐𝑅𝑏))
45	13, 44	mtbid 327	. . . . . . . . 9 ⊢ (𝜑 → ¬ ∃𝑏 ∈ 𝐴 ∀𝑐 ∈ (𝐹 “ 𝑇)𝑐𝑅𝑏)
46		ralnex 3088	. . . . . . . . 9 ⊢ (∀𝑏 ∈ 𝐴 ¬ ∀𝑐 ∈ (𝐹 “ 𝑇)𝑐𝑅𝑏 ↔ ¬ ∃𝑏 ∈ 𝐴 ∀𝑐 ∈ (𝐹 “ 𝑇)𝑐𝑅𝑏)
47	45, 46	sylibr 237	. . . . . . . 8 ⊢ (𝜑 → ∀𝑏 ∈ 𝐴 ¬ ∀𝑐 ∈ (𝐹 “ 𝑇)𝑐𝑅𝑏)
48	47	r19.21bi 3254	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → ¬ ∀𝑐 ∈ (𝐹 “ 𝑇)𝑐𝑅𝑏)
49	18	rneqd 5922	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝑂 = ran (𝐹 ↾ 𝑇))
50		df-ima 5668	. . . . . . . . . . 11 ⊢ (𝐹 “ 𝑇) = ran (𝐹 ↾ 𝑇)
51	49, 50	eqtr4di 2813	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝑂 = (𝐹 “ 𝑇))
52	51	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → ran 𝑂 = (𝐹 “ 𝑇))
53	52	raleqdv 3319	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹 “ 𝑇)𝑐𝑅𝑏))
54	9	ffund 6707	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝑂)
55	54	funfnd 6564	. . . . . . . . . 10 ⊢ (𝜑 → 𝑂 Fn dom 𝑂)
56	55	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → 𝑂 Fn dom 𝑂)
57		breq1 5106	. . . . . . . . . 10 ⊢ (𝑐 = (𝑂‘𝑚) → (𝑐𝑅𝑏 ↔ (𝑂‘𝑚)𝑅𝑏))
58	57	ralrn 7081	. . . . . . . . 9 ⊢ (𝑂 Fn dom 𝑂 → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏))
59	56, 58	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏))
60	53, 59	bitr3d 284	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (∀𝑐 ∈ (𝐹 “ 𝑇)𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏))
61	48, 60	mtbid 327	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → ¬ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏)
62		rexnal 3114	. . . . . 6 ⊢ (∃𝑚 ∈ dom 𝑂 ¬ (𝑂‘𝑚)𝑅𝑏 ↔ ¬ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏)
63	61, 62	sylibr 237	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → ∃𝑚 ∈ dom 𝑂 ¬ (𝑂‘𝑚)𝑅𝑏)
64	1, 2, 3, 4, 5, 6, 7	ordtypelem7 9496	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑚 ∈ dom 𝑂) → ((𝑂‘𝑚)𝑅𝑏 ∨ 𝑏 ∈ ran 𝑂))
65	64	ord 878	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑚 ∈ dom 𝑂) → (¬ (𝑂‘𝑚)𝑅𝑏 → 𝑏 ∈ ran 𝑂))
66	65	rexlimdva 3163	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (∃𝑚 ∈ dom 𝑂 ¬ (𝑂‘𝑚)𝑅𝑏 → 𝑏 ∈ ran 𝑂))
67	63, 66	mpd 16	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ ran 𝑂)
68	10, 67	eqelssd 3952	. . 3 ⊢ (𝜑 → ran 𝑂 = 𝐴)
69		isoeq5 7322	. . 3 ⊢ (ran 𝑂 = 𝐴 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)))
70	68, 69	syl 18	. 2 ⊢ (𝜑 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)))
71	8, 70	mpbid 235	1 ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ∩ cin 3898   class class class wbr 5103   ↦ cmpt 5186   E cep 5554   Se wse 5606   We wwe 5607  dom cdm 5655  ran crn 5656   ↾ cres 5657   “ cima 5658  Ord word 6356  Oncon0 6357  Lim wlim 6358  Fun wfun 6527   Fn wfn 6528  ‘cfv 6533   Isom wiso 6534  ℩crio 7369  recscrecs 8359  OrdIsocoi 9481

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 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

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 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 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 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7370  df-ov 7416  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-oi 9482

This theorem is used by:  ordtypelem10  9499  ordtype2  9506