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Theorem nfoi 7245
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 7241 . 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 4385 . . . 4  |-  F/ x  R  We  A
52, 3nfse 4384 . . . 4  |-  F/ x  R Se  A
64, 5nfan 1783 . . 3  |-  F/ x
( R  We  A  /\  R Se  A )
7 nfcv 2432 . . . . . 6  |-  F/_ x _V
8 nfcv 2432 . . . . . . . . . 10  |-  F/_ x ran  h
9 nfcv 2432 . . . . . . . . . . 11  |-  F/_ x
j
10 nfcv 2432 . . . . . . . . . . 11  |-  F/_ x w
119, 2, 10nfbr 4083 . . . . . . . . . 10  |-  F/ x  j R w
128, 11nfral 2609 . . . . . . . . 9  |-  F/ x A. j  e.  ran  h  j R w
1312, 3nfrab 2734 . . . . . . . 8  |-  F/_ x { w  e.  A  |  A. j  e.  ran  h  j R w }
14 nfcv 2432 . . . . . . . . . 10  |-  F/_ x u
15 nfcv 2432 . . . . . . . . . 10  |-  F/_ x
v
1614, 2, 15nfbr 4083 . . . . . . . . 9  |-  F/ x  u R v
1716nfn 1777 . . . . . . . 8  |-  F/ x  -.  u R v
1813, 17nfral 2609 . . . . . . 7  |-  F/ x A. u  e.  { w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v
1918, 13nfriota 6330 . . . . . 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 4124 . . . . 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 6406 . . . 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 2432 . . . . . . . 8  |-  F/_ x
a
2321, 22nfima 5036 . . . . . . 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 2432 . . . . . . . 8  |-  F/_ x
z
25 nfcv 2432 . . . . . . . 8  |-  F/_ x
t
2624, 2, 25nfbr 4083 . . . . . . 7  |-  F/ x  z R t
2723, 26nfral 2609 . . . . . 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 2611 . . . . 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 2432 . . . . 5  |-  F/_ x On
3028, 29nfrab 2734 . . . 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 4973 . . 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 2432 . . 3  |-  F/_ x (/)
336, 31, 32nfif 3602 . 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 2429 1  |-  F/_ xOrdIso ( R ,  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358   F/_wnfc 2419   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801   (/)c0 3468   ifcif 3578   class class class wbr 4039    e. cmpt 4093   Se wse 4366    We wwe 4367   Oncon0 4408   ran crn 4706    |` cres 4707   "cima 4708   iota_crio 6313  recscrecs 6403  OrdIsocoi 7240
This theorem is referenced by:  hsmexlem2  8069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279  df-riota 6320  df-recs 6404  df-oi 7241
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