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Theorem smoel2 5949
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 5026 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
21eleq2d 2123 . . . . 5  |-  ( F  Fn  A  ->  ( B  e.  dom  F  <->  B  e.  A ) )
32anbi1d 446 . . . 4  |-  ( F  Fn  A  ->  (
( B  e.  dom  F  /\  C  e.  B
)  <->  ( B  e.  A  /\  C  e.  B ) ) )
43biimprd 151 . . 3  |-  ( F  Fn  A  ->  (
( B  e.  A  /\  C  e.  B
)  ->  ( B  e.  dom  F  /\  C  e.  B ) ) )
5 smoel 5946 . . . 4  |-  ( ( Smo  F  /\  B  e.  dom  F  /\  C  e.  B )  ->  ( F `  C )  e.  ( F `  B
) )
653expib 1118 . . 3  |-  ( Smo 
F  ->  ( ( B  e.  dom  F  /\  C  e.  B )  ->  ( F `  C
)  e.  ( F `
 B ) ) )
74, 6sylan9 395 . 2  |-  ( ( F  Fn  A  /\  Smo  F )  ->  (
( B  e.  A  /\  C  e.  B
)  ->  ( F `  C )  e.  ( F `  B ) ) )
87imp 119 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 101    e. wcel 1409   dom cdm 4373    Fn wfn 4925   ` cfv 4930   Smo wsmo 5931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-tr 3883  df-iord 4131  df-iota 4895  df-fn 4933  df-fv 4938  df-smo 5932
This theorem is referenced by: (None)
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