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Theorem nfoi 9509
Description: Hypothesis builder for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
nfoi.1 𝑥𝑅
nfoi.2 𝑥𝐴
Assertion
Ref Expression
nfoi 𝑥OrdIso(𝑅, 𝐴)

Proof of Theorem nfoi
Dummy variables 𝑎 𝑗 𝑡 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-oi 9505 . 2 OrdIso(𝑅, 𝐴) = if((𝑅 We 𝐴𝑅 Se 𝐴), (recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) ↾ {𝑎 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)𝑧𝑅𝑡}), ∅)
2 nfoi.1 . . . . 5 𝑥𝑅
3 nfoi.2 . . . . 5 𝑥𝐴
42, 3nfwe 5653 . . . 4 𝑥 𝑅 We 𝐴
52, 3nfse 5652 . . . 4 𝑥 𝑅 Se 𝐴
64, 5nfan 1903 . . 3 𝑥(𝑅 We 𝐴𝑅 Se 𝐴)
7 nfcv 2904 . . . . . 6 𝑥V
8 nfcv 2904 . . . . . . . . . 10 𝑥ran
9 nfcv 2904 . . . . . . . . . . 11 𝑥𝑗
10 nfcv 2904 . . . . . . . . . . 11 𝑥𝑤
119, 2, 10nfbr 5196 . . . . . . . . . 10 𝑥 𝑗𝑅𝑤
128, 11nfralw 3309 . . . . . . . . 9 𝑥𝑗 ∈ ran 𝑗𝑅𝑤
1312, 3nfrabw 3469 . . . . . . . 8 𝑥{𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
14 nfcv 2904 . . . . . . . . . 10 𝑥𝑢
15 nfcv 2904 . . . . . . . . . 10 𝑥𝑣
1614, 2, 15nfbr 5196 . . . . . . . . 9 𝑥 𝑢𝑅𝑣
1716nfn 1861 . . . . . . . 8 𝑥 ¬ 𝑢𝑅𝑣
1813, 17nfralw 3309 . . . . . . 7 𝑥𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣
1918, 13nfriota 7378 . . . . . 6 𝑥(𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)
207, 19nfmpt 5256 . . . . 5 𝑥( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
2120nfrecs 8375 . . . 4 𝑥recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)))
22 nfcv 2904 . . . . . . . 8 𝑥𝑎
2321, 22nfima 6068 . . . . . . 7 𝑥(recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)
24 nfcv 2904 . . . . . . . 8 𝑥𝑧
25 nfcv 2904 . . . . . . . 8 𝑥𝑡
2624, 2, 25nfbr 5196 . . . . . . 7 𝑥 𝑧𝑅𝑡
2723, 26nfralw 3309 . . . . . 6 𝑥𝑧 ∈ (recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)𝑧𝑅𝑡
283, 27nfrexw 3311 . . . . 5 𝑥𝑡𝐴𝑧 ∈ (recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)𝑧𝑅𝑡
29 nfcv 2904 . . . . 5 𝑥On
3028, 29nfrabw 3469 . . . 4 𝑥{𝑎 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)𝑧𝑅𝑡}
3121, 30nfres 5984 . . 3 𝑥(recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) ↾ {𝑎 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)𝑧𝑅𝑡})
32 nfcv 2904 . . 3 𝑥
336, 31, 32nfif 4559 . 2 𝑥if((𝑅 We 𝐴𝑅 Se 𝐴), (recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) ↾ {𝑎 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)𝑧𝑅𝑡}), ∅)
341, 33nfcxfr 2902 1 𝑥OrdIso(𝑅, 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397  wnfc 2884  wral 3062  wrex 3071  {crab 3433  Vcvv 3475  c0 4323  ifcif 4529   class class class wbr 5149  cmpt 5232   Se wse 5630   We wwe 5631  ran crn 5678  cres 5679  cima 5680  Oncon0 6365  crio 7364  recscrecs 8370  OrdIsocoi 9504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-iota 6496  df-fv 6552  df-riota 7365  df-ov 7412  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-oi 9505
This theorem is referenced by:  hsmexlem2  10422
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