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Theorem ordtypelem8 8374
Description: Lemma for ordtype 8381. (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 8370 . . . . 5 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
9 fdm 6008 . . . . 5 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
108, 9syl 17 . . . 4 (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
11 inss1 3811 . . . . 5 (𝑇 ∩ dom 𝐹) ⊆ 𝑇
121, 2, 3, 4, 5, 6, 7ordtypelem2 8368 . . . . . 6 (𝜑 → Ord 𝑇)
13 ordsson 6936 . . . . . 6 (Ord 𝑇𝑇 ⊆ On)
1412, 13syl 17 . . . . 5 (𝜑𝑇 ⊆ On)
1511, 14syl5ss 3594 . . . 4 (𝜑 → (𝑇 ∩ dom 𝐹) ⊆ On)
1610, 15eqsstrd 3618 . . 3 (𝜑 → dom 𝑂 ⊆ On)
17 epweon 6930 . . . 4 E We On
18 weso 5065 . . . 4 ( E We On → E Or On)
1917, 18ax-mp 5 . . 3 E Or On
20 soss 5013 . . 3 (dom 𝑂 ⊆ On → ( E Or On → E Or dom 𝑂))
2116, 19, 20mpisyl 21 . 2 (𝜑 → E Or dom 𝑂)
22 frn 6010 . . . . 5 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → ran 𝑂𝐴)
238, 22syl 17 . . . 4 (𝜑 → ran 𝑂𝐴)
24 wess 5061 . . . 4 (ran 𝑂𝐴 → (𝑅 We 𝐴𝑅 We ran 𝑂))
2523, 6, 24sylc 65 . . 3 (𝜑𝑅 We ran 𝑂)
26 weso 5065 . . 3 (𝑅 We ran 𝑂𝑅 Or ran 𝑂)
27 sopo 5012 . . 3 (𝑅 Or ran 𝑂𝑅 Po ran 𝑂)
2825, 26, 273syl 18 . 2 (𝜑𝑅 Po ran 𝑂)
29 ffun 6005 . . . 4 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → Fun 𝑂)
308, 29syl 17 . . 3 (𝜑 → Fun 𝑂)
31 funforn 6079 . . 3 (Fun 𝑂𝑂:dom 𝑂onto→ran 𝑂)
3230, 31sylib 208 . 2 (𝜑𝑂:dom 𝑂onto→ran 𝑂)
33 epel 4988 . . . . 5 (𝑎 E 𝑏𝑎𝑏)
341, 2, 3, 4, 5, 6, 7ordtypelem6 8372 . . . . 5 ((𝜑𝑏 ∈ dom 𝑂) → (𝑎𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3533, 34syl5bi 232 . . . 4 ((𝜑𝑏 ∈ dom 𝑂) → (𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3635ralrimiva 2960 . . 3 (𝜑 → ∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3736ralrimivw 2961 . 2 (𝜑 → ∀𝑎 ∈ dom 𝑂𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
38 soisoi 6532 . 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 1324 1 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  cin 3554  wss 3555   class class class wbr 4613  cmpt 4673   E cep 4983   Po wpo 4993   Or wor 4994   Se wse 5031   We wwe 5032  dom cdm 5074  ran crn 5075  cima 5077  Ord word 5681  Oncon0 5682  Fun wfun 5841  wf 5843  ontowfo 5845  cfv 5847   Isom wiso 5848  crio 6564  recscrecs 7412  OrdIsocoi 8358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-wrecs 7352  df-recs 7413  df-oi 8359
This theorem is referenced by:  ordtypelem9  8375  ordtypelem10  8376  oiiso2  8380
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