Theorem ordtypelem8 8586

Description: Lemma for ordtype 8593. (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 8582	. . . . 5 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
9		fdm 6191	. . . . 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 3981	. . . . 5 ⊢ (𝑇 ∩ dom 𝐹) ⊆ 𝑇
12	1, 2, 3, 4, 5, 6, 7	ordtypelem2 8580	. . . . . 6 ⊢ (𝜑 → Ord 𝑇)
13		ordsson 7136	. . . . . 6 ⊢ (Ord 𝑇 → 𝑇 ⊆ On)
14	12, 13	syl 17	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ On)
15	11, 14	syl5ss 3763	. . . 4 ⊢ (𝜑 → (𝑇 ∩ dom 𝐹) ⊆ On)
16	10, 15	eqsstrd 3788	. . 3 ⊢ (𝜑 → dom 𝑂 ⊆ On)
17		epweon 7130	. . . 4 ⊢ E We On
18		weso 5240	. . . 4 ⊢ ( E We On → E Or On)
19	17, 18	ax-mp 5	. . 3 ⊢ E Or On
20		soss 5188	. . 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 6193	. . . . 5 ⊢ (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → ran 𝑂 ⊆ 𝐴)
23	8, 22	syl 17	. . . 4 ⊢ (𝜑 → ran 𝑂 ⊆ 𝐴)
24		wess 5236	. . . 4 ⊢ (ran 𝑂 ⊆ 𝐴 → (𝑅 We 𝐴 → 𝑅 We ran 𝑂))
25	23, 6, 24	sylc 65	. . 3 ⊢ (𝜑 → 𝑅 We ran 𝑂)
26		weso 5240	. . 3 ⊢ (𝑅 We ran 𝑂 → 𝑅 Or ran 𝑂)
27		sopo 5187	. . 3 ⊢ (𝑅 Or ran 𝑂 → 𝑅 Po ran 𝑂)
28	25, 26, 27	3syl 18	. 2 ⊢ (𝜑 → 𝑅 Po ran 𝑂)
29		ffun 6188	. . . 4 ⊢ (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → Fun 𝑂)
30	8, 29	syl 17	. . 3 ⊢ (𝜑 → Fun 𝑂)
31		funforn 6263	. . 3 ⊢ (Fun 𝑂 ↔ 𝑂:dom 𝑂–onto→ran 𝑂)
32	30, 31	sylib 208	. 2 ⊢ (𝜑 → 𝑂:dom 𝑂–onto→ran 𝑂)
33		epel 5165	. . . . 5 ⊢ (𝑎 E 𝑏 ↔ 𝑎 ∈ 𝑏)
34	1, 2, 3, 4, 5, 6, 7	ordtypelem6 8584	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 ∈ 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))
35	33, 34	syl5bi 232	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))
36	35	ralrimiva 3115	. . 3 ⊢ (𝜑 → ∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))
37	36	ralrimivw 3116	. 2 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑂∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))
38		soisoi 6721	. 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 1477	1 ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ∃wrex 3062  {crab 3065  Vcvv 3351   ∩ cin 3722   ⊆ wss 3723   class class class wbr 4786   ↦ cmpt 4863   E cep 5161   Po wpo 5168   Or wor 5169   Se wse 5206   We wwe 5207  dom cdm 5249  ran crn 5250   “ cima 5252  Ord word 5865  Oncon0 5866  Fun wfun 6025  ⟶wf 6027  –onto→wfo 6029  ‘cfv 6031   Isom wiso 6032  ℩crio 6753  recscrecs 7620  OrdIsocoi 8570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-wrecs 7559  df-recs 7621  df-oi 8571

This theorem is referenced by:  ordtypelem9  8587  ordtypelem10  8588  oiiso2  8592