Theorem ordtypelem8 8313

Description: Lemma for ordtype 8320. (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 8309	. . . . 5 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
9		fdm 5964	. . . . 5 ⊢ (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
11		inss1 3795	. . . . 5 ⊢ (𝑇 ∩ dom 𝐹) ⊆ 𝑇
12	1, 2, 3, 4, 5, 6, 7	ordtypelem2 8307	. . . . . 6 ⊢ (𝜑 → Ord 𝑇)
13		ordsson 6881	. . . . . 6 ⊢ (Ord 𝑇 → 𝑇 ⊆ On)
14	12, 13	syl 17	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ On)
15	11, 14	syl5ss 3579	. . . 4 ⊢ (𝜑 → (𝑇 ∩ dom 𝐹) ⊆ On)
16	10, 15	eqsstrd 3602	. . 3 ⊢ (𝜑 → dom 𝑂 ⊆ On)
17		epweon 6875	. . . 4 ⊢ E We On
18		weso 5029	. . . 4 ⊢ ( E We On → E Or On)
19	17, 18	ax-mp 5	. . 3 ⊢ E Or On
20		soss 4977	. . 3 ⊢ (dom 𝑂 ⊆ On → ( E Or On → E Or dom 𝑂))
21	16, 19, 20	mpisyl 21	. 2 ⊢ (𝜑 → E Or dom 𝑂)
22		frn 5966	. . . . 5 ⊢ (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → ran 𝑂 ⊆ 𝐴)
23	8, 22	syl 17	. . . 4 ⊢ (𝜑 → ran 𝑂 ⊆ 𝐴)
24		wess 5025	. . . 4 ⊢ (ran 𝑂 ⊆ 𝐴 → (𝑅 We 𝐴 → 𝑅 We ran 𝑂))
25	23, 6, 24	sylc 63	. . 3 ⊢ (𝜑 → 𝑅 We ran 𝑂)
26		weso 5029	. . 3 ⊢ (𝑅 We ran 𝑂 → 𝑅 Or ran 𝑂)
27		sopo 4976	. . 3 ⊢ (𝑅 Or ran 𝑂 → 𝑅 Po ran 𝑂)
28	25, 26, 27	3syl 18	. 2 ⊢ (𝜑 → 𝑅 Po ran 𝑂)
29		ffun 5961	. . . 4 ⊢ (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → Fun 𝑂)
30	8, 29	syl 17	. . 3 ⊢ (𝜑 → Fun 𝑂)
31		funforn 6035	. . 3 ⊢ (Fun 𝑂 ↔ 𝑂:dom 𝑂–onto→ran 𝑂)
32	30, 31	sylib 207	. 2 ⊢ (𝜑 → 𝑂:dom 𝑂–onto→ran 𝑂)
33		epel 4952	. . . . 5 ⊢ (𝑎 E 𝑏 ↔ 𝑎 ∈ 𝑏)
34	1, 2, 3, 4, 5, 6, 7	ordtypelem6 8311	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 ∈ 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))
35	33, 34	syl5bi 231	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))
36	35	ralrimiva 2949	. . 3 ⊢ (𝜑 → ∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))
37	36	ralrimivw 2950	. 2 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑂∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))
38		soisoi 6478	. 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))
39	21, 28, 32, 37, 38	syl22anc 1319	1 ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643   E cep 4947   Po wpo 4957   Or wor 4958   Se wse 4995   We wwe 4996  dom cdm 5038  ran crn 5039   “ cima 5041  Ord word 5639  Oncon0 5640  Fun wfun 5798  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804   Isom wiso 5805  ℩crio 6510  recscrecs 7354  OrdIsocoi 8297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-wrecs 7294  df-recs 7355  df-oi 8298

This theorem is referenced by:  ordtypelem9  8314  ordtypelem10  8315  oiiso2  8319