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Theorem ordtypelem8 9550
Description: Lemma for ordtype 9557. (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 9546 . . . . 5 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
98fdmd 6733 . . . 4 (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
10 inss1 4227 . . . . 5 (𝑇 ∩ dom 𝐹) ⊆ 𝑇
111, 2, 3, 4, 5, 6, 7ordtypelem2 9544 . . . . . 6 (𝜑 → Ord 𝑇)
12 ordsson 7786 . . . . . 6 (Ord 𝑇𝑇 ⊆ On)
1311, 12syl 17 . . . . 5 (𝜑𝑇 ⊆ On)
1410, 13sstrid 3988 . . . 4 (𝜑 → (𝑇 ∩ dom 𝐹) ⊆ On)
159, 14eqsstrd 4015 . . 3 (𝜑 → dom 𝑂 ⊆ On)
16 epweon 7778 . . . 4 E We On
17 weso 5669 . . . 4 ( E We On → E Or On)
1816, 17ax-mp 5 . . 3 E Or On
19 soss 5610 . . 3 (dom 𝑂 ⊆ On → ( E Or On → E Or dom 𝑂))
2015, 18, 19mpisyl 21 . 2 (𝜑 → E Or dom 𝑂)
218frnd 6731 . . . 4 (𝜑 → ran 𝑂𝐴)
22 wess 5665 . . . 4 (ran 𝑂𝐴 → (𝑅 We 𝐴𝑅 We ran 𝑂))
2321, 6, 22sylc 65 . . 3 (𝜑𝑅 We ran 𝑂)
24 weso 5669 . . 3 (𝑅 We ran 𝑂𝑅 Or ran 𝑂)
25 sopo 5609 . . 3 (𝑅 Or ran 𝑂𝑅 Po ran 𝑂)
2623, 24, 253syl 18 . 2 (𝜑𝑅 Po ran 𝑂)
278ffund 6727 . . 3 (𝜑 → Fun 𝑂)
28 funforn 6817 . . 3 (Fun 𝑂𝑂:dom 𝑂onto→ran 𝑂)
2927, 28sylib 217 . 2 (𝜑𝑂:dom 𝑂onto→ran 𝑂)
30 epel 5585 . . . . 5 (𝑎 E 𝑏𝑎𝑏)
311, 2, 3, 4, 5, 6, 7ordtypelem6 9548 . . . . 5 ((𝜑𝑏 ∈ dom 𝑂) → (𝑎𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3230, 31biimtrid 241 . . . 4 ((𝜑𝑏 ∈ dom 𝑂) → (𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3332ralrimiva 3135 . . 3 (𝜑 → ∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3433ralrimivw 3139 . 2 (𝜑 → ∀𝑎 ∈ dom 𝑂𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
35 soisoi 7335 . 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 394   = wceq 1533  wcel 2098  wral 3050  wrex 3059  {crab 3418  Vcvv 3461  cin 3943  wss 3944   class class class wbr 5149  cmpt 5232   E cep 5581   Po wpo 5588   Or wor 5589   Se wse 5631   We wwe 5632  dom cdm 5678  ran crn 5679  cima 5681  Ord word 6370  Oncon0 6371  Fun wfun 6543  ontowfo 6547  cfv 6549   Isom wiso 6550  crio 7374  recscrecs 8391  OrdIsocoi 9534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-oi 9535
This theorem is referenced by:  ordtypelem9  9551  ordtypelem10  9552  oiiso2  9556
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