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Theorem ordtypelem8 8977
Description: Lemma for ordtype 8984. (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 8973 . . . . 5 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
98fdmd 6504 . . . 4 (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
10 inss1 4179 . . . . 5 (𝑇 ∩ dom 𝐹) ⊆ 𝑇
111, 2, 3, 4, 5, 6, 7ordtypelem2 8971 . . . . . 6 (𝜑 → Ord 𝑇)
12 ordsson 7489 . . . . . 6 (Ord 𝑇𝑇 ⊆ On)
1311, 12syl 17 . . . . 5 (𝜑𝑇 ⊆ On)
1410, 13sstrid 3953 . . . 4 (𝜑 → (𝑇 ∩ dom 𝐹) ⊆ On)
159, 14eqsstrd 3980 . . 3 (𝜑 → dom 𝑂 ⊆ On)
16 epweon 7482 . . . 4 E We On
17 weso 5523 . . . 4 ( E We On → E Or On)
1816, 17ax-mp 5 . . 3 E Or On
19 soss 5470 . . 3 (dom 𝑂 ⊆ On → ( E Or On → E Or dom 𝑂))
2015, 18, 19mpisyl 21 . 2 (𝜑 → E Or dom 𝑂)
218frnd 6501 . . . 4 (𝜑 → ran 𝑂𝐴)
22 wess 5519 . . . 4 (ran 𝑂𝐴 → (𝑅 We 𝐴𝑅 We ran 𝑂))
2321, 6, 22sylc 65 . . 3 (𝜑𝑅 We ran 𝑂)
24 weso 5523 . . 3 (𝑅 We ran 𝑂𝑅 Or ran 𝑂)
25 sopo 5469 . . 3 (𝑅 Or ran 𝑂𝑅 Po ran 𝑂)
2623, 24, 253syl 18 . 2 (𝜑𝑅 Po ran 𝑂)
278ffund 6498 . . 3 (𝜑 → Fun 𝑂)
28 funforn 6579 . . 3 (Fun 𝑂𝑂:dom 𝑂onto→ran 𝑂)
2927, 28sylib 221 . 2 (𝜑𝑂:dom 𝑂onto→ran 𝑂)
30 epel 5446 . . . . 5 (𝑎 E 𝑏𝑎𝑏)
311, 2, 3, 4, 5, 6, 7ordtypelem6 8975 . . . . 5 ((𝜑𝑏 ∈ dom 𝑂) → (𝑎𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3230, 31syl5bi 245 . . . 4 ((𝜑𝑏 ∈ dom 𝑂) → (𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3332ralrimiva 3174 . . 3 (𝜑 → ∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3433ralrimivw 3175 . 2 (𝜑 → ∀𝑎 ∈ dom 𝑂𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
35 soisoi 7065 . 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 837 1 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2114  wral 3130  wrex 3131  {crab 3134  Vcvv 3469  cin 3907  wss 3908   class class class wbr 5042  cmpt 5122   E cep 5441   Po wpo 5449   Or wor 5450   Se wse 5489   We wwe 5490  dom cdm 5532  ran crn 5533  cima 5535  Ord word 6168  Oncon0 6169  Fun wfun 6328  ontowfo 6332  cfv 6334   Isom wiso 6335  crio 7097  recscrecs 7994  OrdIsocoi 8961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-wrecs 7934  df-recs 7995  df-oi 8962
This theorem is referenced by:  ordtypelem9  8978  ordtypelem10  8979  oiiso2  8983
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