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Theorem oicl 8394
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 2621 . . . . 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 2621 . . . . 5 {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
3 eqid 2621 . . . . 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 8382 . . . 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 2621 . . . 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 473 . . . 4 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 We 𝐴)
8 simpr 477 . . . 4 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 Se 𝐴)
94, 2, 3, 5, 6, 7, 8ordtypelem5 8387 . . 3 ((𝑅 We 𝐴𝑅 Se 𝐴) → (Ord dom 𝐹𝐹:dom 𝐹𝐴))
109simpld 475 . 2 ((𝑅 We 𝐴𝑅 Se 𝐴) → Ord dom 𝐹)
11 ord0 5746 . . 3 Ord ∅
126oi0 8393 . . . . . 6 (¬ (𝑅 We 𝐴𝑅 Se 𝐴) → 𝐹 = ∅)
1312dmeqd 5296 . . . . 5 (¬ (𝑅 We 𝐴𝑅 Se 𝐴) → dom 𝐹 = dom ∅)
14 dm0 5309 . . . . 5 dom ∅ = ∅
1513, 14syl6eq 2671 . . . 4 (¬ (𝑅 We 𝐴𝑅 Se 𝐴) → dom 𝐹 = ∅)
16 ordeq 5699 . . . 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 384   = wceq 1480  wral 2908  wrex 2909  {crab 2912  Vcvv 3190  c0 3897   class class class wbr 4623  cmpt 4683   Se wse 5041   We wwe 5042  dom cdm 5084  ran crn 5085  cima 5087  Ord word 5691  Oncon0 5692  wf 5853  crio 6575  recscrecs 7427  OrdIsocoi 8374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-wrecs 7367  df-recs 7428  df-oi 8375
This theorem is referenced by:  oion  8401  oieu  8404  oismo  8405  oiid  8406  wofib  8410  cantnflt  8529  cantnfp1lem3  8537  cantnflem1b  8543  cantnflem1  8546  wemapwe  8554  cnfcomlem  8556  cnfcom  8557  cnfcom2lem  8558  infxpenlem  8796  hsmexlem1  9208  fpwwe2lem8  9419  fpwwe2lem9  9420  fpwwe2lem10  9421
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