Theorem ordtypelem10 8599

Description: Lemma for ordtype 8604. Using ax-rep 4923, exclude the possibility that 𝑂 is a proper class and does not enumerate all of 𝐴. (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
ordtypelem10	⊢ (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)

Proof of Theorem ordtypelem10

Dummy variables 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 8597	. 2 ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
9	1, 2, 3, 4, 5, 6, 7	ordtypelem4 8593	. . . . 5 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
10		frn 6214	. . . . 5 ⊢ (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → ran 𝑂 ⊆ 𝐴)
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → ran 𝑂 ⊆ 𝐴)
12		simprl 811	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑏 ∈ 𝐴)
13	6	adantr 472	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑅 We 𝐴)
14	7	adantr 472	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑅 Se 𝐴)
15	1, 2, 3, 4, 5, 13, 14	ordtypelem8 8597	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
16		isof1o 6737	. . . . . . . . . . . . 13 ⊢ (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) → 𝑂:dom 𝑂–1-1-onto→ran 𝑂)
17		f1of 6299	. . . . . . . . . . . . 13 ⊢ (𝑂:dom 𝑂–1-1-onto→ran 𝑂 → 𝑂:dom 𝑂⟶ran 𝑂)
18	15, 16, 17	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂:dom 𝑂⟶ran 𝑂)
19		f1of1 6298	. . . . . . . . . . . . . 14 ⊢ (𝑂:dom 𝑂–1-1-onto→ran 𝑂 → 𝑂:dom 𝑂–1-1→ran 𝑂)
20	15, 16, 19	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂:dom 𝑂–1-1→ran 𝑂)
21		simpl 474	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂) → 𝑏 ∈ 𝐴)
22		seex 5229	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 Se 𝐴 ∧ 𝑏 ∈ 𝐴) → {𝑐 ∈ 𝐴 ∣ 𝑐𝑅𝑏} ∈ V)
23	7, 21, 22	syl2an 495	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → {𝑐 ∈ 𝐴 ∣ 𝑐𝑅𝑏} ∈ V)
24	11	adantr 472	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ⊆ 𝐴)
25		rexnal 3133	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑚 ∈ dom 𝑂 ¬ (𝑂‘𝑚)𝑅𝑏 ↔ ¬ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏)
26	1, 2, 3, 4, 5, 6, 7	ordtypelem7 8596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑚 ∈ dom 𝑂) → ((𝑂‘𝑚)𝑅𝑏 ∨ 𝑏 ∈ ran 𝑂))
27	26	ord 391	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑚 ∈ dom 𝑂) → (¬ (𝑂‘𝑚)𝑅𝑏 → 𝑏 ∈ ran 𝑂))
28	27	rexlimdva 3169	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (∃𝑚 ∈ dom 𝑂 ¬ (𝑂‘𝑚)𝑅𝑏 → 𝑏 ∈ ran 𝑂))
29	25, 28	syl5bir 233	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (¬ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏 → 𝑏 ∈ ran 𝑂))
30	29	con1d 139	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (¬ 𝑏 ∈ ran 𝑂 → ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏))
31	30	impr 650	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏)
32		ffun 6209	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → Fun 𝑂)
33	9, 32	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Fun 𝑂)
34		funfn 6079	. . . . . . . . . . . . . . . . . . 19 ⊢ (Fun 𝑂 ↔ 𝑂 Fn dom 𝑂)
35	33, 34	sylib 208	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑂 Fn dom 𝑂)
36	35	adantr 472	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Fn dom 𝑂)
37		breq1 4807	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = (𝑂‘𝑚) → (𝑐𝑅𝑏 ↔ (𝑂‘𝑚)𝑅𝑏))
38	37	ralrn 6526	. . . . . . . . . . . . . . . . 17 ⊢ (𝑂 Fn dom 𝑂 → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏))
39	36, 38	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏))
40	31, 39	mpbird 247	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏)
41		ssrab 3821	. . . . . . . . . . . . . . 15 ⊢ (ran 𝑂 ⊆ {𝑐 ∈ 𝐴 ∣ 𝑐𝑅𝑏} ↔ (ran 𝑂 ⊆ 𝐴 ∧ ∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏))
42	24, 40, 41	sylanbrc 701	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ⊆ {𝑐 ∈ 𝐴 ∣ 𝑐𝑅𝑏})
43	23, 42	ssexd 4957	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ∈ V)
44		f1dmex 7302	. . . . . . . . . . . . 13 ⊢ ((𝑂:dom 𝑂–1-1→ran 𝑂 ∧ ran 𝑂 ∈ V) → dom 𝑂 ∈ V)
45	20, 43, 44	syl2anc 696	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → dom 𝑂 ∈ V)
46		fex 6654	. . . . . . . . . . . 12 ⊢ ((𝑂:dom 𝑂⟶ran 𝑂 ∧ dom 𝑂 ∈ V) → 𝑂 ∈ V)
47	18, 45, 46	syl2anc 696	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 ∈ V)
48	1, 2, 3, 4, 5, 13, 14, 47	ordtypelem9 8598	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
49		isof1o 6737	. . . . . . . . . 10 ⊢ (𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴) → 𝑂:dom 𝑂–1-1-onto→𝐴)
50		f1ofo 6306	. . . . . . . . . 10 ⊢ (𝑂:dom 𝑂–1-1-onto→𝐴 → 𝑂:dom 𝑂–onto→𝐴)
51		forn 6280	. . . . . . . . . 10 ⊢ (𝑂:dom 𝑂–onto→𝐴 → ran 𝑂 = 𝐴)
52	48, 49, 50, 51	4syl 19	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 = 𝐴)
53	12, 52	eleqtrrd 2842	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑏 ∈ ran 𝑂)
54	53	expr 644	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (¬ 𝑏 ∈ ran 𝑂 → 𝑏 ∈ ran 𝑂))
55	54	pm2.18d 124	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ ran 𝑂)
56	55	ex 449	. . . . 5 ⊢ (𝜑 → (𝑏 ∈ 𝐴 → 𝑏 ∈ ran 𝑂))
57	56	ssrdv 3750	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ran 𝑂)
58	11, 57	eqssd 3761	. . 3 ⊢ (𝜑 → ran 𝑂 = 𝐴)
59		isoeq5 6735	. . 3 ⊢ (ran 𝑂 = 𝐴 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)))
60	58, 59	syl 17	. 2 ⊢ (𝜑 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)))
61	8, 60	mpbid 222	1 ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051  {crab 3054  Vcvv 3340   ∩ cin 3714   ⊆ wss 3715   class class class wbr 4804   ↦ cmpt 4881   E cep 5178   Se wse 5223   We wwe 5224  dom cdm 5266  ran crn 5267   “ cima 5269  Oncon0 5884  Fun wfun 6043   Fn wfn 6044  ⟶wf 6045  –1-1→wf1 6046  –onto→wfo 6047  –1-1-onto→wf1o 6048  ‘cfv 6049   Isom wiso 6050  ℩crio 6774  recscrecs 7637  OrdIsocoi 8581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-wrecs 7577  df-recs 7638  df-oi 8582

This theorem is referenced by:  ordtype  8604