MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfoi Structured version   Visualization version   GIF version

Theorem nfoi 8771
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 8767 . 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 5379 . . . 4 𝑥 𝑅 We 𝐴
52, 3nfse 5378 . . . 4 𝑥 𝑅 Se 𝐴
64, 5nfan 1862 . . 3 𝑥(𝑅 We 𝐴𝑅 Se 𝐴)
7 nfcv 2926 . . . . . 6 𝑥V
8 nfcv 2926 . . . . . . . . . 10 𝑥ran
9 nfcv 2926 . . . . . . . . . . 11 𝑥𝑗
10 nfcv 2926 . . . . . . . . . . 11 𝑥𝑤
119, 2, 10nfbr 4972 . . . . . . . . . 10 𝑥 𝑗𝑅𝑤
128, 11nfral 3168 . . . . . . . . 9 𝑥𝑗 ∈ ran 𝑗𝑅𝑤
1312, 3nfrab 3319 . . . . . . . 8 𝑥{𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
14 nfcv 2926 . . . . . . . . . 10 𝑥𝑢
15 nfcv 2926 . . . . . . . . . 10 𝑥𝑣
1614, 2, 15nfbr 4972 . . . . . . . . 9 𝑥 𝑢𝑅𝑣
1716nfn 1819 . . . . . . . 8 𝑥 ¬ 𝑢𝑅𝑣
1813, 17nfral 3168 . . . . . . 7 𝑥𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣
1918, 13nfriota 6944 . . . . . 6 𝑥(𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)
207, 19nfmpt 5020 . . . . 5 𝑥( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
2120nfrecs 7813 . . . 4 𝑥recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)))
22 nfcv 2926 . . . . . . . 8 𝑥𝑎
2321, 22nfima 5775 . . . . . . 7 𝑥(recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)
24 nfcv 2926 . . . . . . . 8 𝑥𝑧
25 nfcv 2926 . . . . . . . 8 𝑥𝑡
2624, 2, 25nfbr 4972 . . . . . . 7 𝑥 𝑧𝑅𝑡
2723, 26nfral 3168 . . . . . 6 𝑥𝑧 ∈ (recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)𝑧𝑅𝑡
283, 27nfrex 3247 . . . . 5 𝑥𝑡𝐴𝑧 ∈ (recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)𝑧𝑅𝑡
29 nfcv 2926 . . . . 5 𝑥On
3028, 29nfrab 3319 . . . 4 𝑥{𝑎 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑎)𝑧𝑅𝑡}
3121, 30nfres 5694 . . 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 2926 . . 3 𝑥
336, 31, 32nfif 4373 . 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 2924 1 𝑥OrdIso(𝑅, 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 387  wnfc 2910  wral 3082  wrex 3083  {crab 3086  Vcvv 3409  c0 4172  ifcif 4344   class class class wbr 4925  cmpt 5004   Se wse 5360   We wwe 5361  ran crn 5404  cres 5405  cima 5406  Oncon0 6026  crio 6934  recscrecs 7809  OrdIsocoi 8766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-po 5322  df-so 5323  df-fr 5362  df-se 5363  df-we 5364  df-xp 5409  df-cnv 5411  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-iota 6149  df-fv 6193  df-riota 6935  df-wrecs 7748  df-recs 7810  df-oi 8767
This theorem is referenced by:  hsmexlem2  9645
  Copyright terms: Public domain W3C validator