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Theorem oicl 8475
Description: The order type of the well-order 𝑅 on 𝐴 is an ordinal. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
Hypothesis
Ref Expression
oicl.1 𝐹 = OrdIso(𝑅, 𝐴)
Assertion
Ref Expression
oicl Ord dom 𝐹

Proof of Theorem oicl
Dummy variables 𝑢 𝑡 𝑣 𝑥 𝑗 𝑤 𝑧 𝑓 𝑖 𝑟 𝑠 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2651 . . . . 5 recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) = recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)))
2 eqid 2651 . . . . 5 {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
3 eqid 2651 . . . . 5 ( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)) = ( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
41, 2, 3ordtypecbv 8463 . . . 4 recs((𝑓 ∈ V ↦ (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))) = recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)))
5 eqid 2651 . . . 4 {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (recs((𝑓 ∈ V ↦ (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))) “ 𝑥)𝑧𝑅𝑡} = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (recs((𝑓 ∈ V ↦ (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))) “ 𝑥)𝑧𝑅𝑡}
6 oicl.1 . . . 4 𝐹 = OrdIso(𝑅, 𝐴)
7 simpl 472 . . . 4 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 We 𝐴)
8 simpr 476 . . . 4 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 Se 𝐴)
94, 2, 3, 5, 6, 7, 8ordtypelem5 8468 . . 3 ((𝑅 We 𝐴𝑅 Se 𝐴) → (Ord dom 𝐹𝐹:dom 𝐹𝐴))
109simpld 474 . 2 ((𝑅 We 𝐴𝑅 Se 𝐴) → Ord dom 𝐹)
11 ord0 5815 . . 3 Ord ∅
126oi0 8474 . . . . . 6 (¬ (𝑅 We 𝐴𝑅 Se 𝐴) → 𝐹 = ∅)
1312dmeqd 5358 . . . . 5 (¬ (𝑅 We 𝐴𝑅 Se 𝐴) → dom 𝐹 = dom ∅)
14 dm0 5371 . . . . 5 dom ∅ = ∅
1513, 14syl6eq 2701 . . . 4 (¬ (𝑅 We 𝐴𝑅 Se 𝐴) → dom 𝐹 = ∅)
16 ordeq 5768 . . . 4 (dom 𝐹 = ∅ → (Ord dom 𝐹 ↔ Ord ∅))
1715, 16syl 17 . . 3 (¬ (𝑅 We 𝐴𝑅 Se 𝐴) → (Ord dom 𝐹 ↔ Ord ∅))
1811, 17mpbiri 248 . 2 (¬ (𝑅 We 𝐴𝑅 Se 𝐴) → Ord dom 𝐹)
1910, 18pm2.61i 176 1 Ord dom 𝐹
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 383   = wceq 1523  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  c0 3948   class class class wbr 4685  cmpt 4762   Se wse 5100   We wwe 5101  dom cdm 5143  ran crn 5144  cima 5146  Ord word 5760  Oncon0 5761  wf 5922  crio 6650  recscrecs 7512  OrdIsocoi 8455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-wrecs 7452  df-recs 7513  df-oi 8456
This theorem is referenced by:  oion  8482  oieu  8485  oismo  8486  oiid  8487  wofib  8491  cantnflt  8607  cantnfp1lem3  8615  cantnflem1b  8621  cantnflem1  8624  wemapwe  8632  cnfcomlem  8634  cnfcom  8635  cnfcom2lem  8636  infxpenlem  8874  hsmexlem1  9286  fpwwe2lem8  9497  fpwwe2lem9  9498  fpwwe2lem10  9499
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