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Theorem heiborlem2 26521
Description: Lemma for heibor 26530. 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 2496 . . 3  |-  ( y  =  A  ->  (
y  e.  ( F `
 n )  <->  A  e.  ( F `  n ) ) )
4 oveq1 6088 . . . 4  |-  ( y  =  A  ->  (
y B n )  =  ( A B n ) )
54eleq1d 2502 . . 3  |-  ( y  =  A  ->  (
( y B n )  e.  K  <->  ( A B n )  e.  K ) )
63, 53anbi23d 1257 . 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 2496 . . 3  |-  ( n  =  C  ->  (
n  e.  NN0  <->  C  e.  NN0 ) )
8 fveq2 5728 . . . 4  |-  ( n  =  C  ->  ( F `  n )  =  ( F `  C ) )
98eleq2d 2503 . . 3  |-  ( n  =  C  ->  ( A  e.  ( F `  n )  <->  A  e.  ( F `  C ) ) )
10 oveq2 6089 . . . 4  |-  ( n  =  C  ->  ( A B n )  =  ( A B C ) )
1110eleq1d 2502 . . 3  |-  ( n  =  C  ->  (
( A B n )  e.  K  <->  ( A B C )  e.  K
) )
127, 9, 113anbi123d 1254 . 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 4477 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 177    /\ w3a 936    = wceq 1652    e. wcel 1725   {cab 2422   E.wrex 2706   _Vcvv 2956    i^i cin 3319    C_ wss 3320   ~Pcpw 3799   U.cuni 4015   class class class wbr 4212   {copab 4265   ` cfv 5454  (class class class)co 6081   Fincfn 7109   NN0cn0 10221   MetOpencmopn 16691
This theorem is referenced by:  heiborlem3  26522  heiborlem5  26524  heiborlem6  26525  heiborlem8  26527  heiborlem10  26529
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-iota 5418  df-fv 5462  df-ov 6084
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