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Theorem oicl 9562
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 2726 . . . . 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 2726 . . . . 5 {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
3 eqid 2726 . . . . 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 9550 . . . 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 2726 . . . 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 481 . . . 4 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 We 𝐴)
8 simpr 483 . . . 4 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 Se 𝐴)
94, 2, 3, 5, 6, 7, 8ordtypelem5 9555 . . 3 ((𝑅 We 𝐴𝑅 Se 𝐴) → (Ord dom 𝐹𝐹:dom 𝐹𝐴))
109simpld 493 . 2 ((𝑅 We 𝐴𝑅 Se 𝐴) → Ord dom 𝐹)
11 ord0 6418 . . 3 Ord ∅
126oi0 9561 . . . . . 6 (¬ (𝑅 We 𝐴𝑅 Se 𝐴) → 𝐹 = ∅)
1312dmeqd 5902 . . . . 5 (¬ (𝑅 We 𝐴𝑅 Se 𝐴) → dom 𝐹 = dom ∅)
14 dm0 5917 . . . . 5 dom ∅ = ∅
1513, 14eqtrdi 2782 . . . 4 (¬ (𝑅 We 𝐴𝑅 Se 𝐴) → dom 𝐹 = ∅)
16 ordeq 6372 . . . 4 (dom 𝐹 = ∅ → (Ord dom 𝐹 ↔ Ord ∅))
1715, 16syl 17 . . 3 (¬ (𝑅 We 𝐴𝑅 Se 𝐴) → (Ord dom 𝐹 ↔ Ord ∅))
1811, 17mpbiri 257 . 2 (¬ (𝑅 We 𝐴𝑅 Se 𝐴) → Ord dom 𝐹)
1910, 18pm2.61i 182 1 Ord dom 𝐹
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 394   = wceq 1534  wral 3051  wrex 3060  {crab 3419  Vcvv 3462  c0 4322   class class class wbr 5143  cmpt 5226   Se wse 5625   We wwe 5626  dom cdm 5672  ran crn 5673  cima 5675  Ord word 6364  Oncon0 6365  wf 6539  crio 7368  recscrecs 8389  OrdIsocoi 9542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6302  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-oi 9543
This theorem is referenced by:  oion  9569  oieu  9572  oismo  9573  oiid  9574  wofib  9578  cantnflt  9705  cantnfp1lem3  9713  cantnflem1b  9719  cantnflem1  9722  wemapwe  9730  cnfcomlem  9732  cnfcom  9733  cnfcom2lem  9734  infxpenlem  10046  hsmexlem1  10457  fpwwe2lem7  10668  fpwwe2lem8  10669  fpwwe2lem9  10670
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