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Theorem smoel2 7026
Description: A strictly monotone ordinal function preserves the epsilon relation. (Contributed by Mario Carneiro, 12-Mar-2013.)
Assertion
Ref Expression
smoel2  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( B  e.  A  /\  C  e.  B
) )  ->  ( F `  C )  e.  ( F `  B
) )

Proof of Theorem smoel2
StepHypRef Expression
1 fndm 5662 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
21eleq2d 2524 . . . . 5  |-  ( F  Fn  A  ->  ( B  e.  dom  F  <->  B  e.  A ) )
32anbi1d 702 . . . 4  |-  ( F  Fn  A  ->  (
( B  e.  dom  F  /\  C  e.  B
)  <->  ( B  e.  A  /\  C  e.  B ) ) )
43biimprd 223 . . 3  |-  ( F  Fn  A  ->  (
( B  e.  A  /\  C  e.  B
)  ->  ( B  e.  dom  F  /\  C  e.  B ) ) )
5 smoel 7023 . . . 4  |-  ( ( Smo  F  /\  B  e.  dom  F  /\  C  e.  B )  ->  ( F `  C )  e.  ( F `  B
) )
653expib 1197 . . 3  |-  ( Smo 
F  ->  ( ( B  e.  dom  F  /\  C  e.  B )  ->  ( F `  C
)  e.  ( F `
 B ) ) )
74, 6sylan9 655 . 2  |-  ( ( F  Fn  A  /\  Smo  F )  ->  (
( B  e.  A  /\  C  e.  B
)  ->  ( F `  C )  e.  ( F `  B ) ) )
87imp 427 1  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( B  e.  A  /\  C  e.  B
) )  ->  ( F `  C )  e.  ( F `  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1823   dom cdm 4988    Fn wfn 5565   ` cfv 5570   Smo wsmo 7008
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-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-tr 4533  df-ord 4870  df-iota 5534  df-fn 5573  df-fv 5578  df-smo 7009
This theorem is referenced by:  smo11  7027  smoord  7028  smogt  7030  cofsmo  8640
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