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Theorem heiborlem2 25684
Description: Lemma for heibor 25693. Substitutions for the set  G. (Contributed by Jeff Madsen, 23-Jan-2014.)
Hypotheses
Ref Expression
heibor.1  |-  J  =  ( MetOpen `  D )
heibor.3  |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin )
u  C_  U. v }
heibor.4  |-  G  =  { <. y ,  n >.  |  ( n  e. 
NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K
) }
heiborlem2.5  |-  A  e. 
_V
heiborlem2.6  |-  C  e. 
_V
Assertion
Ref Expression
heiborlem2  |-  ( A G C  <->  ( C  e.  NN0  /\  A  e.  ( F `  C
)  /\  ( A B C )  e.  K
) )
Distinct variable groups:    y, n, A    u, n, F, y   
v, n, D, u, y    B, n, u, v, y    n, J, u, v, y    U, n, u, v, y    C, n, u, v, y    n, K, y
Allowed substitution hints:    A( v, u)    F( v)    G( y, v, u, n)    K( v, u)

Proof of Theorem heiborlem2
StepHypRef Expression
1 heiborlem2.5 . 2  |-  A  e. 
_V
2 heiborlem2.6 . 2  |-  C  e. 
_V
3 eleq1 2376 . . 3  |-  ( y  =  A  ->  (
y  e.  ( F `
 n )  <->  A  e.  ( F `  n ) ) )
4 oveq1 5907 . . . 4  |-  ( y  =  A  ->  (
y B n )  =  ( A B n ) )
54eleq1d 2382 . . 3  |-  ( y  =  A  ->  (
( y B n )  e.  K  <->  ( A B n )  e.  K ) )
63, 53anbi23d 1255 . 2  |-  ( y  =  A  ->  (
( n  e.  NN0  /\  y  e.  ( F `
 n )  /\  ( y B n )  e.  K )  <-> 
( n  e.  NN0  /\  A  e.  ( F `
 n )  /\  ( A B n )  e.  K ) ) )
7 eleq1 2376 . . 3  |-  ( n  =  C  ->  (
n  e.  NN0  <->  C  e.  NN0 ) )
8 fveq2 5563 . . . 4  |-  ( n  =  C  ->  ( F `  n )  =  ( F `  C ) )
98eleq2d 2383 . . 3  |-  ( n  =  C  ->  ( A  e.  ( F `  n )  <->  A  e.  ( F `  C ) ) )
10 oveq2 5908 . . . 4  |-  ( n  =  C  ->  ( A B n )  =  ( A B C ) )
1110eleq1d 2382 . . 3  |-  ( n  =  C  ->  (
( A B n )  e.  K  <->  ( A B C )  e.  K
) )
127, 9, 113anbi123d 1252 . 2  |-  ( n  =  C  ->  (
( n  e.  NN0  /\  A  e.  ( F `
 n )  /\  ( A B n )  e.  K )  <->  ( C  e.  NN0  /\  A  e.  ( F `  C
)  /\  ( A B C )  e.  K
) ) )
13 heibor.4 . 2  |-  G  =  { <. y ,  n >.  |  ( n  e. 
NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K
) }
141, 2, 6, 12, 13brab 4324 1  |-  ( A G C  <->  ( C  e.  NN0  /\  A  e.  ( F `  C
)  /\  ( A B C )  e.  K
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ w3a 934    = wceq 1633    e. wcel 1701   {cab 2302   E.wrex 2578   _Vcvv 2822    i^i cin 3185    C_ wss 3186   ~Pcpw 3659   U.cuni 3864   class class class wbr 4060   {copab 4113   ` cfv 5292  (class class class)co 5900   Fincfn 6906   NN0cn0 10012   MetOpencmopn 16423
This theorem is referenced by:  heiborlem3  25685  heiborlem5  25687  heiborlem6  25688  heiborlem8  25690  heiborlem10  25692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-iota 5256  df-fv 5300  df-ov 5903
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