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Theorem ordtypelem8 8786
Description: Lemma for ordtype 8793. (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 8782 . . . . 5 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
98fdmd 6355 . . . 4 (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
10 inss1 4094 . . . . 5 (𝑇 ∩ dom 𝐹) ⊆ 𝑇
111, 2, 3, 4, 5, 6, 7ordtypelem2 8780 . . . . . 6 (𝜑 → Ord 𝑇)
12 ordsson 7322 . . . . . 6 (Ord 𝑇𝑇 ⊆ On)
1311, 12syl 17 . . . . 5 (𝜑𝑇 ⊆ On)
1410, 13syl5ss 3871 . . . 4 (𝜑 → (𝑇 ∩ dom 𝐹) ⊆ On)
159, 14eqsstrd 3897 . . 3 (𝜑 → dom 𝑂 ⊆ On)
16 epweon 7315 . . . 4 E We On
17 weso 5399 . . . 4 ( E We On → E Or On)
1816, 17ax-mp 5 . . 3 E Or On
19 soss 5346 . . 3 (dom 𝑂 ⊆ On → ( E Or On → E Or dom 𝑂))
2015, 18, 19mpisyl 21 . 2 (𝜑 → E Or dom 𝑂)
218frnd 6353 . . . 4 (𝜑 → ran 𝑂𝐴)
22 wess 5395 . . . 4 (ran 𝑂𝐴 → (𝑅 We 𝐴𝑅 We ran 𝑂))
2321, 6, 22sylc 65 . . 3 (𝜑𝑅 We ran 𝑂)
24 weso 5399 . . 3 (𝑅 We ran 𝑂𝑅 Or ran 𝑂)
25 sopo 5345 . . 3 (𝑅 Or ran 𝑂𝑅 Po ran 𝑂)
2623, 24, 253syl 18 . 2 (𝜑𝑅 Po ran 𝑂)
278ffund 6350 . . 3 (𝜑 → Fun 𝑂)
28 funforn 6428 . . 3 (Fun 𝑂𝑂:dom 𝑂onto→ran 𝑂)
2927, 28sylib 210 . 2 (𝜑𝑂:dom 𝑂onto→ran 𝑂)
30 epel 5322 . . . . 5 (𝑎 E 𝑏𝑎𝑏)
311, 2, 3, 4, 5, 6, 7ordtypelem6 8784 . . . . 5 ((𝜑𝑏 ∈ dom 𝑂) → (𝑎𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3230, 31syl5bi 234 . . . 4 ((𝜑𝑏 ∈ dom 𝑂) → (𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3332ralrimiva 3132 . . 3 (𝜑 → ∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3433ralrimivw 3133 . 2 (𝜑 → ∀𝑎 ∈ dom 𝑂𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
35 soisoi 6906 . 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))
3620, 26, 29, 34, 35syl22anc 826 1 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387   = wceq 1507  wcel 2050  wral 3088  wrex 3089  {crab 3092  Vcvv 3415  cin 3830  wss 3831   class class class wbr 4930  cmpt 5009   E cep 5317   Po wpo 5325   Or wor 5326   Se wse 5365   We wwe 5366  dom cdm 5408  ran crn 5409  cima 5411  Ord word 6030  Oncon0 6031  Fun wfun 6184  ontowfo 6188  cfv 6190   Isom wiso 6191  crio 6938  recscrecs 7813  OrdIsocoi 8770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-se 5368  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-isom 6199  df-riota 6939  df-wrecs 7752  df-recs 7814  df-oi 8771
This theorem is referenced by:  ordtypelem9  8787  ordtypelem10  8788  oiiso2  8792
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