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Theorem ordtypelem8 8593
Description: Lemma for ordtype 8600. (Contributed by Mario Carneiro, 25-Jun-2015.)
Hypotheses
Ref Expression
ordtypelem.1 𝐹 = recs(𝐺)
ordtypelem.2 𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
ordtypelem.3 𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))
ordtypelem.5 𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}
ordtypelem.6 𝑂 = OrdIso(𝑅, 𝐴)
ordtypelem.7 (𝜑𝑅 We 𝐴)
ordtypelem.8 (𝜑𝑅 Se 𝐴)
Assertion
Ref Expression
ordtypelem8 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
Distinct variable groups:   𝑣,𝑢,𝐶   ,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧,𝑅   𝐴,,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧   𝑡,𝑂,𝑢,𝑣,𝑥   𝜑,𝑡,𝑥   ,𝐹,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑧,𝑤,𝑣,𝑢,,𝑗)   𝐶(𝑥,𝑧,𝑤,𝑡,,𝑗)   𝑇(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,,𝑗)   𝐺(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,,𝑗)   𝑂(𝑧,𝑤,,𝑗)

Proof of Theorem ordtypelem8
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtypelem.1 . . . . . 6 𝐹 = recs(𝐺)
2 ordtypelem.2 . . . . . 6 𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
3 ordtypelem.3 . . . . . 6 𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))
4 ordtypelem.5 . . . . . 6 𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}
5 ordtypelem.6 . . . . . 6 𝑂 = OrdIso(𝑅, 𝐴)
6 ordtypelem.7 . . . . . 6 (𝜑𝑅 We 𝐴)
7 ordtypelem.8 . . . . . 6 (𝜑𝑅 Se 𝐴)
81, 2, 3, 4, 5, 6, 7ordtypelem4 8589 . . . . 5 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
9 fdm 6210 . . . . 5 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
108, 9syl 17 . . . 4 (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
11 inss1 3974 . . . . 5 (𝑇 ∩ dom 𝐹) ⊆ 𝑇
121, 2, 3, 4, 5, 6, 7ordtypelem2 8587 . . . . . 6 (𝜑 → Ord 𝑇)
13 ordsson 7152 . . . . . 6 (Ord 𝑇𝑇 ⊆ On)
1412, 13syl 17 . . . . 5 (𝜑𝑇 ⊆ On)
1511, 14syl5ss 3753 . . . 4 (𝜑 → (𝑇 ∩ dom 𝐹) ⊆ On)
1610, 15eqsstrd 3778 . . 3 (𝜑 → dom 𝑂 ⊆ On)
17 epweon 7146 . . . 4 E We On
18 weso 5255 . . . 4 ( E We On → E Or On)
1917, 18ax-mp 5 . . 3 E Or On
20 soss 5203 . . 3 (dom 𝑂 ⊆ On → ( E Or On → E Or dom 𝑂))
2116, 19, 20mpisyl 21 . 2 (𝜑 → E Or dom 𝑂)
22 frn 6212 . . . . 5 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → ran 𝑂𝐴)
238, 22syl 17 . . . 4 (𝜑 → ran 𝑂𝐴)
24 wess 5251 . . . 4 (ran 𝑂𝐴 → (𝑅 We 𝐴𝑅 We ran 𝑂))
2523, 6, 24sylc 65 . . 3 (𝜑𝑅 We ran 𝑂)
26 weso 5255 . . 3 (𝑅 We ran 𝑂𝑅 Or ran 𝑂)
27 sopo 5202 . . 3 (𝑅 Or ran 𝑂𝑅 Po ran 𝑂)
2825, 26, 273syl 18 . 2 (𝜑𝑅 Po ran 𝑂)
29 ffun 6207 . . . 4 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → Fun 𝑂)
308, 29syl 17 . . 3 (𝜑 → Fun 𝑂)
31 funforn 6281 . . 3 (Fun 𝑂𝑂:dom 𝑂onto→ran 𝑂)
3230, 31sylib 208 . 2 (𝜑𝑂:dom 𝑂onto→ran 𝑂)
33 epel 5180 . . . . 5 (𝑎 E 𝑏𝑎𝑏)
341, 2, 3, 4, 5, 6, 7ordtypelem6 8591 . . . . 5 ((𝜑𝑏 ∈ dom 𝑂) → (𝑎𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3533, 34syl5bi 232 . . . 4 ((𝜑𝑏 ∈ dom 𝑂) → (𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3635ralrimiva 3102 . . 3 (𝜑 → ∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3736ralrimivw 3103 . 2 (𝜑 → ∀𝑎 ∈ dom 𝑂𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
38 soisoi 6739 . 2 ((( E Or dom 𝑂𝑅 Po ran 𝑂) ∧ (𝑂:dom 𝑂onto→ran 𝑂 ∧ ∀𝑎 ∈ dom 𝑂𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))) → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
3921, 28, 32, 37, 38syl22anc 1478 1 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1630  wcel 2137  wral 3048  wrex 3049  {crab 3052  Vcvv 3338  cin 3712  wss 3713   class class class wbr 4802  cmpt 4879   E cep 5176   Po wpo 5183   Or wor 5184   Se wse 5221   We wwe 5222  dom cdm 5264  ran crn 5265  cima 5267  Ord word 5881  Oncon0 5882  Fun wfun 6041  wf 6043  ontowfo 6045  cfv 6047   Isom wiso 6048  crio 6771  recscrecs 7634  OrdIsocoi 8577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-se 5224  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-isom 6056  df-riota 6772  df-wrecs 7574  df-recs 7635  df-oi 8578
This theorem is referenced by:  ordtypelem9  8594  ordtypelem10  8595  oiiso2  8599
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