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Theorem smoel2 6068
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 5113 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
21eleq2d 2157 . . . . 5  |-  ( F  Fn  A  ->  ( B  e.  dom  F  <->  B  e.  A ) )
32anbi1d 453 . . . 4  |-  ( F  Fn  A  ->  (
( B  e.  dom  F  /\  C  e.  B
)  <->  ( B  e.  A  /\  C  e.  B ) ) )
43biimprd 156 . . 3  |-  ( F  Fn  A  ->  (
( B  e.  A  /\  C  e.  B
)  ->  ( B  e.  dom  F  /\  C  e.  B ) ) )
5 smoel 6065 . . . 4  |-  ( ( Smo  F  /\  B  e.  dom  F  /\  C  e.  B )  ->  ( F `  C )  e.  ( F `  B
) )
653expib 1146 . . 3  |-  ( Smo 
F  ->  ( ( B  e.  dom  F  /\  C  e.  B )  ->  ( F `  C
)  e.  ( F `
 B ) ) )
74, 6sylan9 401 . 2  |-  ( ( F  Fn  A  /\  Smo  F )  ->  (
( B  e.  A  /\  C  e.  B
)  ->  ( F `  C )  e.  ( F `  B ) ) )
87imp 122 1  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( B  e.  A  /\  C  e.  B
) )  ->  ( F `  C )  e.  ( F `  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1438   dom cdm 4438    Fn wfn 5010   ` cfv 5015   Smo wsmo 6050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-tr 3937  df-iord 4193  df-iota 4980  df-fn 5018  df-fv 5023  df-smo 6051
This theorem is referenced by: (None)
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