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Theorem nfoi 7935
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 7931 . 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 4855 . . . 4  |-  F/ x  R  We  A
52, 3nfse 4854 . . . 4  |-  F/ x  R Se  A
64, 5nfan 1875 . . 3  |-  F/ x
( R  We  A  /\  R Se  A )
7 nfcv 2629 . . . . . 6  |-  F/_ x _V
8 nfcv 2629 . . . . . . . . . 10  |-  F/_ x ran  h
9 nfcv 2629 . . . . . . . . . . 11  |-  F/_ x
j
10 nfcv 2629 . . . . . . . . . . 11  |-  F/_ x w
119, 2, 10nfbr 4491 . . . . . . . . . 10  |-  F/ x  j R w
128, 11nfral 2850 . . . . . . . . 9  |-  F/ x A. j  e.  ran  h  j R w
1312, 3nfrab 3043 . . . . . . . 8  |-  F/_ x { w  e.  A  |  A. j  e.  ran  h  j R w }
14 nfcv 2629 . . . . . . . . . 10  |-  F/_ x u
15 nfcv 2629 . . . . . . . . . 10  |-  F/_ x
v
1614, 2, 15nfbr 4491 . . . . . . . . 9  |-  F/ x  u R v
1716nfn 1849 . . . . . . . 8  |-  F/ x  -.  u R v
1813, 17nfral 2850 . . . . . . 7  |-  F/ x A. u  e.  { w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v
1918, 13nfriota 6252 . . . . . 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 4535 . . . . 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 7041 . . . 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 2629 . . . . . . . 8  |-  F/_ x
a
2321, 22nfima 5343 . . . . . . 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 2629 . . . . . . . 8  |-  F/_ x
z
25 nfcv 2629 . . . . . . . 8  |-  F/_ x
t
2624, 2, 25nfbr 4491 . . . . . . 7  |-  F/ x  z R t
2723, 26nfral 2850 . . . . . 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 2927 . . . . 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 2629 . . . . 5  |-  F/_ x On
3028, 29nfrab 3043 . . . 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 5273 . . 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 2629 . . 3  |-  F/_ x (/)
336, 31, 32nfif 3968 . 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 2627 1  |-  F/_ xOrdIso ( R ,  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369   F/_wnfc 2615   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113   (/)c0 3785   ifcif 3939   class class class wbr 4447    |-> cmpt 4505   Se wse 4836    We wwe 4837   Oncon0 4878   ran crn 5000    |` cres 5001   "cima 5002   iota_crio 6242  recscrecs 7038  OrdIsocoi 7930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fv 5594  df-riota 6243  df-recs 7039  df-oi 7931
This theorem is referenced by:  hsmexlem2  8803
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