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Theorem nobndlem3 29694
Description: Lemma for nobndup 29700 and nobnddown 29701. 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 29693 . . 3  |-  ( ( F  C_  No  /\  F  e.  A )  ->  C  e.  On )
41noxpsgn 29665 . . 3  |-  ( C  e.  On  ->  ( C  X.  { X }
)  e.  No )
5 bdayval 29648 . . 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 3116 . . . 4  |-  X  e. 
_V
87snnz 4134 . . 3  |-  { X }  =/=  (/)
9 dmxp 5210 . . 3  |-  ( { X }  =/=  (/)  ->  dom  ( C  X.  { X } )  =  C )
108, 9ax-mp 5 . 2  |-  dom  ( C  X.  { X }
)  =  C
116, 10syl6eq 2511 1  |-  ( ( F  C_  No  /\  F  e.  A )  ->  ( bday `  ( C  X.  { X } ) )  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   {crab 2808    C_ wss 3461   (/)c0 3783   {csn 4016   {cpr 4018   |^|cint 4271   Oncon0 4867    X. cxp 4986   dom cdm 4988   ` cfv 5570   1oc1o 7115   2oc2o 7116   Nocsur 29640   bdaycbday 29642
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-rep 4550  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-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-int 4272  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-we 4829  df-ord 4870  df-on 4871  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-1o 7122  df-2o 7123  df-no 29643  df-bday 29645
This theorem is referenced by:  nobndlem8  29699
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