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Theorem nfoi 7972
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 7968 . 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 4798 . . . 4  |-  F/ x  R  We  A
52, 3nfse 4797 . . . 4  |-  F/ x  R Se  A
64, 5nfan 1956 . . 3  |-  F/ x
( R  We  A  /\  R Se  A )
7 nfcv 2564 . . . . . 6  |-  F/_ x _V
8 nfcv 2564 . . . . . . . . . 10  |-  F/_ x ran  h
9 nfcv 2564 . . . . . . . . . . 11  |-  F/_ x
j
10 nfcv 2564 . . . . . . . . . . 11  |-  F/_ x w
119, 2, 10nfbr 4438 . . . . . . . . . 10  |-  F/ x  j R w
128, 11nfral 2789 . . . . . . . . 9  |-  F/ x A. j  e.  ran  h  j R w
1312, 3nfrab 2988 . . . . . . . 8  |-  F/_ x { w  e.  A  |  A. j  e.  ran  h  j R w }
14 nfcv 2564 . . . . . . . . . 10  |-  F/_ x u
15 nfcv 2564 . . . . . . . . . 10  |-  F/_ x
v
1614, 2, 15nfbr 4438 . . . . . . . . 9  |-  F/ x  u R v
1716nfn 1929 . . . . . . . 8  |-  F/ x  -.  u R v
1813, 17nfral 2789 . . . . . . 7  |-  F/ x A. u  e.  { w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v
1918, 13nfriota 6248 . . . . . 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 4482 . . . . 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 7077 . . . 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 2564 . . . . . . . 8  |-  F/_ x
a
2321, 22nfima 5164 . . . . . . 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 2564 . . . . . . . 8  |-  F/_ x
z
25 nfcv 2564 . . . . . . . 8  |-  F/_ x
t
2624, 2, 25nfbr 4438 . . . . . . 7  |-  F/ x  z R t
2723, 26nfral 2789 . . . . . 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 2866 . . . . 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 2564 . . . . 5  |-  F/_ x On
3028, 29nfrab 2988 . . . 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 5095 . . 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 2564 . . 3  |-  F/_ x (/)
336, 31, 32nfif 3913 . 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 2562 1  |-  F/_ xOrdIso ( R ,  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 367   F/_wnfc 2550   A.wral 2753   E.wrex 2754   {crab 2757   _Vcvv 3058   (/)c0 3737   ifcif 3884   class class class wbr 4394    |-> cmpt 4452   Se wse 4779    We wwe 4780   ran crn 4823    |` cres 4824   "cima 4825   Oncon0 5409   iota_crio 6238  recscrecs 7073  OrdIsocoi 7967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-cnv 4830  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-iota 5532  df-fv 5576  df-riota 6239  df-wrecs 7012  df-recs 7074  df-oi 7968
This theorem is referenced by:  hsmexlem2  8838
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