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Theorem nfoi 9063
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 9059 . 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 5511 . . . 4 𝑥 𝑅 We 𝐴
52, 3nfse 5510 . . . 4 𝑥 𝑅 Se 𝐴
64, 5nfan 1906 . . 3 𝑥(𝑅 We 𝐴𝑅 Se 𝐴)
7 nfcv 2900 . . . . . 6 𝑥V
8 nfcv 2900 . . . . . . . . . 10 𝑥ran
9 nfcv 2900 . . . . . . . . . . 11 𝑥𝑗
10 nfcv 2900 . . . . . . . . . . 11 𝑥𝑤
119, 2, 10nfbr 5087 . . . . . . . . . 10 𝑥 𝑗𝑅𝑤
128, 11nfralw 3139 . . . . . . . . 9 𝑥𝑗 ∈ ran 𝑗𝑅𝑤
1312, 3nfrabw 3289 . . . . . . . 8 𝑥{𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
14 nfcv 2900 . . . . . . . . . 10 𝑥𝑢
15 nfcv 2900 . . . . . . . . . 10 𝑥𝑣
1614, 2, 15nfbr 5087 . . . . . . . . 9 𝑥 𝑢𝑅𝑣
1716nfn 1864 . . . . . . . 8 𝑥 ¬ 𝑢𝑅𝑣
1813, 17nfralw 3139 . . . . . . 7 𝑥𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣
1918, 13nfriota 7152 . . . . . 6 𝑥(𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)
207, 19nfmpt 5137 . . . . 5 𝑥( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
2120nfrecs 8052 . . . 4 𝑥recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)))
22 nfcv 2900 . . . . . . . 8 𝑥𝑎
2321, 22nfima 5921 . . . . . . 7 𝑥(recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)
24 nfcv 2900 . . . . . . . 8 𝑥𝑧
25 nfcv 2900 . . . . . . . 8 𝑥𝑡
2624, 2, 25nfbr 5087 . . . . . . 7 𝑥 𝑧𝑅𝑡
2723, 26nfralw 3139 . . . . . 6 𝑥𝑧 ∈ (recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)𝑧𝑅𝑡
283, 27nfrex 3220 . . . . 5 𝑥𝑡𝐴𝑧 ∈ (recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)𝑧𝑅𝑡
29 nfcv 2900 . . . . 5 𝑥On
3028, 29nfrabw 3289 . . . 4 𝑥{𝑎 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)𝑧𝑅𝑡}
3121, 30nfres 5837 . . 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 2900 . . 3 𝑥
336, 31, 32nfif 4454 . 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 2898 1 𝑥OrdIso(𝑅, 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399  wnfc 2880  wral 3054  wrex 3055  {crab 3058  Vcvv 3400  c0 4221  ifcif 4424   class class class wbr 5040  cmpt 5120   Se wse 5491   We wwe 5492  ran crn 5536  cres 5537  cima 5538  Oncon0 6182  crio 7138  recscrecs 8048  OrdIsocoi 9058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-mpt 5121  df-po 5452  df-so 5453  df-fr 5493  df-se 5494  df-we 5495  df-xp 5541  df-cnv 5543  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-pred 6139  df-iota 6307  df-fv 6357  df-riota 7139  df-wrecs 7988  df-recs 8049  df-oi 9059
This theorem is referenced by:  hsmexlem2  9939
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