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Theorem nfoi 8038
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 8034 . 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 4829 . . . 4  |-  F/ x  R  We  A
52, 3nfse 4828 . . . 4  |-  F/ x  R Se  A
64, 5nfan 1988 . . 3  |-  F/ x
( R  We  A  /\  R Se  A )
7 nfcv 2580 . . . . . 6  |-  F/_ x _V
8 nfcv 2580 . . . . . . . . . 10  |-  F/_ x ran  h
9 nfcv 2580 . . . . . . . . . . 11  |-  F/_ x
j
10 nfcv 2580 . . . . . . . . . . 11  |-  F/_ x w
119, 2, 10nfbr 4468 . . . . . . . . . 10  |-  F/ x  j R w
128, 11nfral 2808 . . . . . . . . 9  |-  F/ x A. j  e.  ran  h  j R w
1312, 3nfrab 3007 . . . . . . . 8  |-  F/_ x { w  e.  A  |  A. j  e.  ran  h  j R w }
14 nfcv 2580 . . . . . . . . . 10  |-  F/_ x u
15 nfcv 2580 . . . . . . . . . 10  |-  F/_ x
v
1614, 2, 15nfbr 4468 . . . . . . . . 9  |-  F/ x  u R v
1716nfn 1960 . . . . . . . 8  |-  F/ x  -.  u R v
1813, 17nfral 2808 . . . . . . 7  |-  F/ x A. u  e.  { w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v
1918, 13nfriota 6276 . . . . . 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 4512 . . . . 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 7104 . . . 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 2580 . . . . . . . 8  |-  F/_ x
a
2321, 22nfima 5195 . . . . . . 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 2580 . . . . . . . 8  |-  F/_ x
z
25 nfcv 2580 . . . . . . . 8  |-  F/_ x
t
2624, 2, 25nfbr 4468 . . . . . . 7  |-  F/ x  z R t
2723, 26nfral 2808 . . . . . 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 2885 . . . . 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 2580 . . . . 5  |-  F/_ x On
3028, 29nfrab 3007 . . . 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 5126 . . 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 2580 . . 3  |-  F/_ x (/)
336, 31, 32nfif 3940 . 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 2578 1  |-  F/_ xOrdIso ( R ,  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 370   F/_wnfc 2566   A.wral 2771   E.wrex 2772   {crab 2775   _Vcvv 3080   (/)c0 3761   ifcif 3911   class class class wbr 4423    |-> cmpt 4482   Se wse 4810    We wwe 4811   ran crn 4854    |` cres 4855   "cima 4856   Oncon0 5442   iota_crio 6266  recscrecs 7100  OrdIsocoi 8033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-cnv 4861  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-iota 5565  df-fv 5609  df-riota 6267  df-wrecs 7039  df-recs 7101  df-oi 8034
This theorem is referenced by:  hsmexlem2  8864
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