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Theorem nfoi 8363
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 8359 . 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 5050 . . . 4 𝑥 𝑅 We 𝐴
52, 3nfse 5049 . . . 4 𝑥 𝑅 Se 𝐴
64, 5nfan 1825 . . 3 𝑥(𝑅 We 𝐴𝑅 Se 𝐴)
7 nfcv 2761 . . . . . 6 𝑥V
8 nfcv 2761 . . . . . . . . . 10 𝑥ran
9 nfcv 2761 . . . . . . . . . . 11 𝑥𝑗
10 nfcv 2761 . . . . . . . . . . 11 𝑥𝑤
119, 2, 10nfbr 4659 . . . . . . . . . 10 𝑥 𝑗𝑅𝑤
128, 11nfral 2940 . . . . . . . . 9 𝑥𝑗 ∈ ran 𝑗𝑅𝑤
1312, 3nfrab 3112 . . . . . . . 8 𝑥{𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
14 nfcv 2761 . . . . . . . . . 10 𝑥𝑢
15 nfcv 2761 . . . . . . . . . 10 𝑥𝑣
1614, 2, 15nfbr 4659 . . . . . . . . 9 𝑥 𝑢𝑅𝑣
1716nfn 1781 . . . . . . . 8 𝑥 ¬ 𝑢𝑅𝑣
1813, 17nfral 2940 . . . . . . 7 𝑥𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣
1918, 13nfriota 6574 . . . . . 6 𝑥(𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)
207, 19nfmpt 4706 . . . . 5 𝑥( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
2120nfrecs 7416 . . . 4 𝑥recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)))
22 nfcv 2761 . . . . . . . 8 𝑥𝑎
2321, 22nfima 5433 . . . . . . 7 𝑥(recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)
24 nfcv 2761 . . . . . . . 8 𝑥𝑧
25 nfcv 2761 . . . . . . . 8 𝑥𝑡
2624, 2, 25nfbr 4659 . . . . . . 7 𝑥 𝑧𝑅𝑡
2723, 26nfral 2940 . . . . . 6 𝑥𝑧 ∈ (recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)𝑧𝑅𝑡
283, 27nfrex 3001 . . . . 5 𝑥𝑡𝐴𝑧 ∈ (recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)𝑧𝑅𝑡
29 nfcv 2761 . . . . 5 𝑥On
3028, 29nfrab 3112 . . . 4 𝑥{𝑎 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)𝑧𝑅𝑡}
3121, 30nfres 5358 . . 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 2761 . . 3 𝑥
336, 31, 32nfif 4087 . 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 2759 1 𝑥OrdIso(𝑅, 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384  wnfc 2748  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  c0 3891  ifcif 4058   class class class wbr 4613  cmpt 4673   Se wse 5031   We wwe 5032  ran crn 5075  cres 5076  cima 5077  Oncon0 5682  crio 6564  recscrecs 7412  OrdIsocoi 8358
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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
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-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-iota 5810  df-fv 5855  df-riota 6565  df-wrecs 7352  df-recs 7413  df-oi 8359
This theorem is referenced by:  hsmexlem2  9193
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