Theorem ordtypelem8 9565

Description: Lemma for ordtype 9572. (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
ordtypelem8	⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))

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 ordtypelem8

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

Step	Hyp	Ref	Expression
1		ordtypelem.1	. . . . . 6 ⊢ 𝐹 = recs(𝐺)
2		ordtypelem.2	. . . . . 6 ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
3		ordtypelem.3	. . . . . 6 ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))
4		ordtypelem.5	. . . . . 6 ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}
5		ordtypelem.6	. . . . . 6 ⊢ 𝑂 = OrdIso(𝑅, 𝐴)
6		ordtypelem.7	. . . . . 6 ⊢ (𝜑 → 𝑅 We 𝐴)
7		ordtypelem.8	. . . . . 6 ⊢ (𝜑 → 𝑅 Se 𝐴)
8	1, 2, 3, 4, 5, 6, 7	ordtypelem4 9561	. . . . 5 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
9	8	fdmd 6746	. . . 4 ⊢ (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
10		inss1 4237	. . . . 5 ⊢ (𝑇 ∩ dom 𝐹) ⊆ 𝑇
11	1, 2, 3, 4, 5, 6, 7	ordtypelem2 9559	. . . . . 6 ⊢ (𝜑 → Ord 𝑇)
12		ordsson 7803	. . . . . 6 ⊢ (Ord 𝑇 → 𝑇 ⊆ On)
13	11, 12	syl 17	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ On)
14	10, 13	sstrid 3995	. . . 4 ⊢ (𝜑 → (𝑇 ∩ dom 𝐹) ⊆ On)
15	9, 14	eqsstrd 4018	. . 3 ⊢ (𝜑 → dom 𝑂 ⊆ On)
16		epweon 7795	. . . 4 ⊢ E We On
17		weso 5676	. . . 4 ⊢ ( E We On → E Or On)
18	16, 17	ax-mp 5	. . 3 ⊢ E Or On
19		soss 5612	. . 3 ⊢ (dom 𝑂 ⊆ On → ( E Or On → E Or dom 𝑂))
20	15, 18, 19	mpisyl 21	. 2 ⊢ (𝜑 → E Or dom 𝑂)
21	8	frnd 6744	. . . 4 ⊢ (𝜑 → ran 𝑂 ⊆ 𝐴)
22		wess 5671	. . . 4 ⊢ (ran 𝑂 ⊆ 𝐴 → (𝑅 We 𝐴 → 𝑅 We ran 𝑂))
23	21, 6, 22	sylc 65	. . 3 ⊢ (𝜑 → 𝑅 We ran 𝑂)
24		weso 5676	. . 3 ⊢ (𝑅 We ran 𝑂 → 𝑅 Or ran 𝑂)
25		sopo 5611	. . 3 ⊢ (𝑅 Or ran 𝑂 → 𝑅 Po ran 𝑂)
26	23, 24, 25	3syl 18	. 2 ⊢ (𝜑 → 𝑅 Po ran 𝑂)
27	8	ffund 6740	. . 3 ⊢ (𝜑 → Fun 𝑂)
28		funforn 6827	. . 3 ⊢ (Fun 𝑂 ↔ 𝑂:dom 𝑂–onto→ran 𝑂)
29	27, 28	sylib 218	. 2 ⊢ (𝜑 → 𝑂:dom 𝑂–onto→ran 𝑂)
30		epel 5587	. . . . 5 ⊢ (𝑎 E 𝑏 ↔ 𝑎 ∈ 𝑏)
31	1, 2, 3, 4, 5, 6, 7	ordtypelem6 9563	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 ∈ 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))
32	30, 31	biimtrid 242	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))
33	32	ralrimiva 3146	. . 3 ⊢ (𝜑 → ∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))
34	33	ralrimivw 3150	. 2 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑂∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))
35		soisoi 7348	. 2 ⊢ ((( E Or dom 𝑂 ∧ 𝑅 Po ran 𝑂) ∧ (𝑂:dom 𝑂–onto→ran 𝑂 ∧ ∀𝑎 ∈ dom 𝑂∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))) → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
36	20, 26, 29, 34, 35	syl22anc 839	1 ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  {crab 3436  Vcvv 3480   ∩ cin 3950   ⊆ wss 3951   class class class wbr 5143   ↦ cmpt 5225   E cep 5583   Po wpo 5590   Or wor 5591   Se wse 5635   We wwe 5636  dom cdm 5685  ran crn 5686   “ cima 5688  Ord word 6383  Oncon0 6384  Fun wfun 6555  –onto→wfo 6559  ‘cfv 6561   Isom wiso 6562  ℩crio 7387  recscrecs 8410  OrdIsocoi 9549

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 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-oi 9550

This theorem is referenced by:  ordtypelem9  9566  ordtypelem10  9567  oiiso2  9571