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Theorem bclelnu 25258
Description: The base class of an element of a nuple. (Contributed by FL, 19-Jun-2011.)
Hypothesis
Ref Expression
bcelnu.1  |-  ( x  =  I  ->  B  =  C )
Assertion
Ref Expression
bclelnu  |-  ( ( F  e.  X_ x  e.  A  B  /\  I  e.  A )  ->  ( F `  I
)  e.  C )
Distinct variable groups:    x, A    x, C    x, F    x, I
Allowed substitution hint:    B( x)

Proof of Theorem bclelnu
StepHypRef Expression
1 elixp2b 25257 . 2  |-  ( F  e.  X_ x  e.  A  B  ->  A. x  e.  A  ( F `  x )  e.  B )
2 fveq2 5541 . . . 4  |-  ( x  =  I  ->  ( F `  x )  =  ( F `  I ) )
3 bcelnu.1 . . . 4  |-  ( x  =  I  ->  B  =  C )
42, 3eleq12d 2364 . . 3  |-  ( x  =  I  ->  (
( F `  x
)  e.  B  <->  ( F `  I )  e.  C
) )
54rspccva 2896 . 2  |-  ( ( A. x  e.  A  ( F `  x )  e.  B  /\  I  e.  A )  ->  ( F `  I )  e.  C )
61, 5sylan 457 1  |-  ( ( F  e.  X_ x  e.  A  B  /\  I  e.  A )  ->  ( F `  I
)  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   ` cfv 5271   X_cixp 6833
This theorem is referenced by:  prmapcp2  25260  cbicp  25269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ixp 6834
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