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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 = {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 8582 . . . . 5 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
9 fdm 6191 . . . . 5 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
108, 9syl 17 . . . 4 (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
11 inss1 3981 . . . . 5 (𝑇 ∩ dom 𝐹) ⊆ 𝑇
121, 2, 3, 4, 5, 6, 7ordtypelem2 8580 . . . . . 6 (𝜑 → Ord 𝑇)
13 ordsson 7136 . . . . . 6 (Ord 𝑇𝑇 ⊆ On)
1412, 13syl 17 . . . . 5 (𝜑𝑇 ⊆ On)
1511, 14syl5ss 3763 . . . 4 (𝜑 → (𝑇 ∩ dom 𝐹) ⊆ On)
1610, 15eqsstrd 3788 . . 3 (𝜑 → dom 𝑂 ⊆ On)
17 epweon 7130 . . . 4 E We On
18 weso 5240 . . . 4 ( E We On → E Or On)
1917, 18ax-mp 5 . . 3 E Or On
20 soss 5188 . . 3 (dom 𝑂 ⊆ On → ( E Or On → E Or dom 𝑂))
2116, 19, 20mpisyl 21 . 2 (𝜑 → E Or dom 𝑂)
22 frn 6193 . . . . 5 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → ran 𝑂𝐴)
238, 22syl 17 . . . 4 (𝜑 → ran 𝑂𝐴)
24 wess 5236 . . . 4 (ran 𝑂𝐴 → (𝑅 We 𝐴𝑅 We ran 𝑂))
2523, 6, 24sylc 65 . . 3 (𝜑𝑅 We ran 𝑂)
26 weso 5240 . . 3 (𝑅 We ran 𝑂𝑅 Or ran 𝑂)
27 sopo 5187 . . 3 (𝑅 Or ran 𝑂𝑅 Po ran 𝑂)
2825, 26, 273syl 18 . 2 (𝜑𝑅 Po ran 𝑂)
29 ffun 6188 . . . 4 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → Fun 𝑂)
308, 29syl 17 . . 3 (𝜑 → Fun 𝑂)
31 funforn 6263 . . 3 (Fun 𝑂𝑂:dom 𝑂onto→ran 𝑂)
3230, 31sylib 208 . 2 (𝜑𝑂:dom 𝑂onto→ran 𝑂)
33 epel 5165 . . . . 5 (𝑎 E 𝑏𝑎𝑏)
341, 2, 3, 4, 5, 6, 7ordtypelem6 8584 . . . . 5 ((𝜑𝑏 ∈ dom 𝑂) → (𝑎𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3533, 34syl5bi 232 . . . 4 ((𝜑𝑏 ∈ dom 𝑂) → (𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3635ralrimiva 3115 . . 3 (𝜑 → ∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3736ralrimivw 3116 . 2 (𝜑 → ∀𝑎 ∈ dom 𝑂𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
38 soisoi 6721 . 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 1477 1 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
Colors of variables: wff setvar class
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  ontowfo 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
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