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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  –onto→wfo 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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