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Theorem oif 8675
Description: The order isomorphism of the well-order 𝑅 on 𝐴 is a function. (Contributed by Mario Carneiro, 23-May-2015.)
Hypothesis
Ref Expression
oicl.1 𝐹 = OrdIso(𝑅, 𝐴)
Assertion
Ref Expression
oif 𝐹:dom 𝐹𝐴

Proof of Theorem oif
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 2797 . . . . 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 2797 . . . . 5 {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
3 eqid 2797 . . . . 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 8662 . . . 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 2797 . . . 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 475 . . . 4 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 We 𝐴)
8 simpr 478 . . . 4 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 Se 𝐴)
94, 2, 3, 5, 6, 7, 8ordtypelem5 8667 . . 3 ((𝑅 We 𝐴𝑅 Se 𝐴) → (Ord dom 𝐹𝐹:dom 𝐹𝐴))
109simprd 490 . 2 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝐹:dom 𝐹𝐴)
11 f0 6299 . . 3 ∅:∅⟶𝐴
126oi0 8673 . . . 4 (¬ (𝑅 We 𝐴𝑅 Se 𝐴) → 𝐹 = ∅)
1312dmeqd 5527 . . . . 5 (¬ (𝑅 We 𝐴𝑅 Se 𝐴) → dom 𝐹 = dom ∅)
14 dm0 5540 . . . . 5 dom ∅ = ∅
1513, 14syl6eq 2847 . . . 4 (¬ (𝑅 We 𝐴𝑅 Se 𝐴) → dom 𝐹 = ∅)
1612, 15feq12d 6242 . . 3 (¬ (𝑅 We 𝐴𝑅 Se 𝐴) → (𝐹:dom 𝐹𝐴 ↔ ∅:∅⟶𝐴))
1711, 16mpbiri 250 . 2 (¬ (𝑅 We 𝐴𝑅 Se 𝐴) → 𝐹:dom 𝐹𝐴)
1810, 17pm2.61i 177 1 𝐹:dom 𝐹𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 385   = wceq 1653  wral 3087  wrex 3088  {crab 3091  Vcvv 3383  c0 4113   class class class wbr 4841  cmpt 4920   Se wse 5267   We wwe 5268  dom cdm 5310  ran crn 5311  cima 5313  Ord word 5938  Oncon0 5939  wf 6095  crio 6836  recscrecs 7704  OrdIsocoi 8654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-se 5270  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-wrecs 7643  df-recs 7705  df-oi 8655
This theorem is referenced by:  oismo  8685  cantnfle  8816  cantnflt  8817  cantnfres  8822  cantnfp1lem3  8825  cantnflem1b  8831  cantnflem1  8834  wemapwe  8842  cnfcomlem  8844  cnfcom  8845  cnfcom3lem  8848  cnfcom3  8849  hsmexlem1  9534  hsmexlem2  9535  fpwwe2lem8  9745
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