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Theorem oiiniseg 7950
Description:  ran  F is an initial segment of  A under the well-order  R. (Contributed by Mario Carneiro, 26-Jun-2015.)
Hypothesis
Ref Expression
oicl.1  |-  F  = OrdIso
( R ,  A
)
Assertion
Ref Expression
oiiniseg  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( N  e.  A  /\  M  e.  dom  F ) )  ->  (
( F `  M
) R N  \/  N  e.  ran  F ) )

Proof of Theorem oiiniseg
Dummy variables  u  t  v  x  h  j  w  z  f 
i  r  s  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2454 . . . 4  |- 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 ) ) )  = 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 ) ) )
2 eqid 2454 . . . 4  |-  { w  e.  A  |  A. j  e.  ran  h  j R w }  =  { w  e.  A  |  A. j  e.  ran  h  j R w }
3 eqid 2454 . . . 4  |-  ( 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 ) )  =  ( 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 ) )
41, 2, 3ordtypecbv 7934 . . 3  |- recs ( ( f  e.  _V  |->  (
iota_ s  e.  { y  e.  A  |  A. i  e.  ran  f  i R y } A. r  e.  { y  e.  A  |  A. i  e.  ran  f  i R y }  -.  r R s ) ) )  = 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 ) ) )
5 eqid 2454 . . 3  |-  { x  e.  On  |  E. t  e.  A  A. z  e.  (recs ( ( f  e.  _V  |->  ( iota_ s  e.  { y  e.  A  |  A. i  e.  ran  f  i R y } A. r  e.  { y  e.  A  |  A. i  e.  ran  f  i R y }  -.  r R s ) ) )
" x ) z R t }  =  { x  e.  On  |  E. t  e.  A  A. z  e.  (recs ( ( f  e. 
_V  |->  ( iota_ s  e. 
{ y  e.  A  |  A. i  e.  ran  f  i R y } A. r  e. 
{ y  e.  A  |  A. i  e.  ran  f  i R y }  -.  r R s ) ) )
" x ) z R t }
6 oicl.1 . . 3  |-  F  = OrdIso
( R ,  A
)
7 simpl 455 . . 3  |-  ( ( R  We  A  /\  R Se  A )  ->  R  We  A )
8 simpr 459 . . 3  |-  ( ( R  We  A  /\  R Se  A )  ->  R Se  A )
94, 2, 3, 5, 6, 7, 8ordtypelem7 7941 . 2  |-  ( ( ( ( R  We  A  /\  R Se  A )  /\  N  e.  A
)  /\  M  e.  dom  F )  ->  (
( F `  M
) R N  \/  N  e.  ran  F ) )
109anasss 645 1  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( N  e.  A  /\  M  e.  dom  F ) )  ->  (
( F `  M
) R N  \/  N  e.  ran  F ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106   class class class wbr 4439    |-> cmpt 4497   Se wse 4825    We wwe 4826   Oncon0 4867   dom cdm 4988   ran crn 4989   "cima 4991   ` cfv 5570   iota_crio 6231  recscrecs 7033  OrdIsocoi 7926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-recs 7034  df-oi 7927
This theorem is referenced by:  oismo  7957
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