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Theorem ordtypelem8 9562
Description: Lemma for ordtype 9569. (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 9558 . . . . 5 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
98fdmd 6746 . . . 4 (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
10 inss1 4244 . . . . 5 (𝑇 ∩ dom 𝐹) ⊆ 𝑇
111, 2, 3, 4, 5, 6, 7ordtypelem2 9556 . . . . . 6 (𝜑 → Ord 𝑇)
12 ordsson 7801 . . . . . 6 (Ord 𝑇𝑇 ⊆ On)
1311, 12syl 17 . . . . 5 (𝜑𝑇 ⊆ On)
1410, 13sstrid 4006 . . . 4 (𝜑 → (𝑇 ∩ dom 𝐹) ⊆ On)
159, 14eqsstrd 4033 . . 3 (𝜑 → dom 𝑂 ⊆ On)
16 epweon 7793 . . . 4 E We On
17 weso 5679 . . . 4 ( E We On → E Or On)
1816, 17ax-mp 5 . . 3 E Or On
19 soss 5616 . . 3 (dom 𝑂 ⊆ On → ( E Or On → E Or dom 𝑂))
2015, 18, 19mpisyl 21 . 2 (𝜑 → E Or dom 𝑂)
218frnd 6744 . . . 4 (𝜑 → ran 𝑂𝐴)
22 wess 5674 . . . 4 (ran 𝑂𝐴 → (𝑅 We 𝐴𝑅 We ran 𝑂))
2321, 6, 22sylc 65 . . 3 (𝜑𝑅 We ran 𝑂)
24 weso 5679 . . 3 (𝑅 We ran 𝑂𝑅 Or ran 𝑂)
25 sopo 5615 . . 3 (𝑅 Or ran 𝑂𝑅 Po ran 𝑂)
2623, 24, 253syl 18 . 2 (𝜑𝑅 Po ran 𝑂)
278ffund 6740 . . 3 (𝜑 → Fun 𝑂)
28 funforn 6827 . . 3 (Fun 𝑂𝑂:dom 𝑂onto→ran 𝑂)
2927, 28sylib 218 . 2 (𝜑𝑂:dom 𝑂onto→ran 𝑂)
30 epel 5591 . . . . 5 (𝑎 E 𝑏𝑎𝑏)
311, 2, 3, 4, 5, 6, 7ordtypelem6 9560 . . . . 5 ((𝜑𝑏 ∈ dom 𝑂) → (𝑎𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3230, 31biimtrid 242 . . . 4 ((𝜑𝑏 ∈ dom 𝑂) → (𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3332ralrimiva 3143 . . 3 (𝜑 → ∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3433ralrimivw 3147 . 2 (𝜑 → ∀𝑎 ∈ dom 𝑂𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
35 soisoi 7347 . 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 839 1 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1536  wcel 2105  wral 3058  wrex 3067  {crab 3432  Vcvv 3477  cin 3961  wss 3962   class class class wbr 5147  cmpt 5230   E cep 5587   Po wpo 5594   Or wor 5595   Se wse 5638   We wwe 5639  dom cdm 5688  ran crn 5689  cima 5691  Ord word 6384  Oncon0 6385  Fun wfun 6556  ontowfo 6560  cfv 6562   Isom wiso 6563  crio 7386  recscrecs 8408  OrdIsocoi 9546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-oi 9547
This theorem is referenced by:  ordtypelem9  9563  ordtypelem10  9564  oiiso2  9568
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