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Theorem nfoi 7740
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  |-  F/_ x R
nfoi.2  |-  F/_ x A
Assertion
Ref Expression
nfoi  |-  F/_ xOrdIso ( R ,  A )

Proof of Theorem nfoi
Dummy variables  h  a  j  t  u  v  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-oi 7736 . 2  |- OrdIso ( R ,  A )  =  if ( ( R  We  A  /\  R Se  A ) ,  (recs ( ( h  e. 
_V  |->  ( iota_ v  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )  |`  { a  e.  On  |  E. t  e.  A  A. z  e.  (recs ( ( h  e. 
_V  |->  ( iota_ v  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )
" a ) z R t } ) ,  (/) )
2 nfoi.1 . . . . 5  |-  F/_ x R
3 nfoi.2 . . . . 5  |-  F/_ x A
42, 3nfwe 4708 . . . 4  |-  F/ x  R  We  A
52, 3nfse 4707 . . . 4  |-  F/ x  R Se  A
64, 5nfan 1861 . . 3  |-  F/ x
( R  We  A  /\  R Se  A )
7 nfcv 2589 . . . . . 6  |-  F/_ x _V
8 nfcv 2589 . . . . . . . . . 10  |-  F/_ x ran  h
9 nfcv 2589 . . . . . . . . . . 11  |-  F/_ x
j
10 nfcv 2589 . . . . . . . . . . 11  |-  F/_ x w
119, 2, 10nfbr 4348 . . . . . . . . . 10  |-  F/ x  j R w
128, 11nfral 2781 . . . . . . . . 9  |-  F/ x A. j  e.  ran  h  j R w
1312, 3nfrab 2914 . . . . . . . 8  |-  F/_ x { w  e.  A  |  A. j  e.  ran  h  j R w }
14 nfcv 2589 . . . . . . . . . 10  |-  F/_ x u
15 nfcv 2589 . . . . . . . . . 10  |-  F/_ x
v
1614, 2, 15nfbr 4348 . . . . . . . . 9  |-  F/ x  u R v
1716nfn 1835 . . . . . . . 8  |-  F/ x  -.  u R v
1813, 17nfral 2781 . . . . . . 7  |-  F/ x A. u  e.  { w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v
1918, 13nfriota 6073 . . . . . 6  |-  F/_ x
( iota_ v  e.  {
w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v )
207, 19nfmpt 4392 . . . . 5  |-  F/_ x
( h  e.  _V  |->  ( iota_ v  e.  {
w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) )
2120nfrecs 6846 . . . 4  |-  F/_ xrecs ( ( h  e. 
_V  |->  ( iota_ v  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )
22 nfcv 2589 . . . . . . . 8  |-  F/_ x
a
2321, 22nfima 5189 . . . . . . 7  |-  F/_ x
(recs ( ( h  e.  _V  |->  ( iota_ v  e.  { w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e.  { w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )
" a )
24 nfcv 2589 . . . . . . . 8  |-  F/_ x
z
25 nfcv 2589 . . . . . . . 8  |-  F/_ x
t
2624, 2, 25nfbr 4348 . . . . . . 7  |-  F/ x  z R t
2723, 26nfral 2781 . . . . . 6  |-  F/ x A. z  e.  (recs ( ( h  e. 
_V  |->  ( iota_ v  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )
" a ) z R t
283, 27nfrex 2783 . . . . 5  |-  F/ x E. t  e.  A  A. z  e.  (recs ( ( h  e. 
_V  |->  ( iota_ v  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )
" a ) z R t
29 nfcv 2589 . . . . 5  |-  F/_ x On
3028, 29nfrab 2914 . . . 4  |-  F/_ x { a  e.  On  |  E. t  e.  A  A. z  e.  (recs ( ( h  e. 
_V  |->  ( iota_ v  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )
" a ) z R t }
3121, 30nfres 5124 . . 3  |-  F/_ x
(recs ( ( h  e.  _V  |->  ( iota_ v  e.  { w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e.  { w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )  |`  { a  e.  On  |  E. t  e.  A  A. z  e.  (recs ( ( h  e. 
_V  |->  ( iota_ v  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )
" a ) z R t } )
32 nfcv 2589 . . 3  |-  F/_ x (/)
336, 31, 32nfif 3830 . 2  |-  F/_ x if ( ( R  We  A  /\  R Se  A ) ,  (recs ( ( h  e.  _V  |->  (
iota_ v  e.  { w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e.  { w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )  |`  { a  e.  On  |  E. t  e.  A  A. z  e.  (recs ( ( h  e.  _V  |->  ( iota_ v  e.  { w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e.  { w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )
" a ) z R t } ) ,  (/) )
341, 33nfcxfr 2587 1  |-  F/_ xOrdIso ( R ,  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369   F/_wnfc 2575   A.wral 2727   E.wrex 2728   {crab 2731   _Vcvv 2984   (/)c0 3649   ifcif 3803   class class class wbr 4304    e. cmpt 4362   Se wse 4689    We wwe 4690   Oncon0 4731   ran crn 4853    |` cres 4854   "cima 4855   iota_crio 6063  recscrecs 6843  OrdIsocoi 7735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-xp 4858  df-cnv 4860  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fv 5438  df-riota 6064  df-recs 6844  df-oi 7736
This theorem is referenced by:  hsmexlem2  8608
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