Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nobndlem3 Structured version   Unicode version

Theorem nobndlem3 29307
Description: Lemma for nobndup 29313 and nobnddown 29314. Calculate the birthday of  ( C  X.  { X } ). (Contributed by Scott Fenton, 17-Aug-2011.)
Hypotheses
Ref Expression
nobndlem2.1  |-  X  e. 
{ 1o ,  2o }
nobndlem2.2  |-  C  = 
|^| { a  e.  On  |  A. n  e.  F  E. b  e.  a 
( n `  b
)  =/=  X }
Assertion
Ref Expression
nobndlem3  |-  ( ( F  C_  No  /\  F  e.  A )  ->  ( bday `  ( C  X.  { X } ) )  =  C )
Distinct variable groups:    F, a,
b, n    X, a,
b
Allowed substitution hints:    A( n, a, b)    C( n, a, b)    X( n)

Proof of Theorem nobndlem3
StepHypRef Expression
1 nobndlem2.1 . . . 4  |-  X  e. 
{ 1o ,  2o }
2 nobndlem2.2 . . . 4  |-  C  = 
|^| { a  e.  On  |  A. n  e.  F  E. b  e.  a 
( n `  b
)  =/=  X }
31, 2nobndlem2 29306 . . 3  |-  ( ( F  C_  No  /\  F  e.  A )  ->  C  e.  On )
41noxpsgn 29278 . . 3  |-  ( C  e.  On  ->  ( C  X.  { X }
)  e.  No )
5 bdayval 29261 . . 3  |-  ( ( C  X.  { X } )  e.  No  ->  ( bday `  ( C  X.  { X }
) )  =  dom  ( C  X.  { X } ) )
63, 4, 53syl 20 . 2  |-  ( ( F  C_  No  /\  F  e.  A )  ->  ( bday `  ( C  X.  { X } ) )  =  dom  ( C  X.  { X }
) )
71elexi 3123 . . . 4  |-  X  e. 
_V
87snnz 4145 . . 3  |-  { X }  =/=  (/)
9 dmxp 5221 . . 3  |-  ( { X }  =/=  (/)  ->  dom  ( C  X.  { X } )  =  C )
108, 9ax-mp 5 . 2  |-  dom  ( C  X.  { X }
)  =  C
116, 10syl6eq 2524 1  |-  ( ( F  C_  No  /\  F  e.  A )  ->  ( bday `  ( C  X.  { X } ) )  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818    C_ wss 3476   (/)c0 3785   {csn 4027   {cpr 4029   |^|cint 4282   Oncon0 4878    X. cxp 4997   dom cdm 4999   ` cfv 5588   1oc1o 7124   2oc2o 7125   Nocsur 29253   bdaycbday 29255
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
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-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-1o 7131  df-2o 7132  df-no 29256  df-bday 29258
This theorem is referenced by:  nobndlem8  29312
  Copyright terms: Public domain W3C validator