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Theorem smoel2 6380
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 5343 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
21eleq2d 2350 . . . . 5  |-  ( F  Fn  A  ->  ( B  e.  dom  F  <->  B  e.  A ) )
32anbi1d 685 . . . 4  |-  ( F  Fn  A  ->  (
( B  e.  dom  F  /\  C  e.  B
)  <->  ( B  e.  A  /\  C  e.  B ) ) )
43biimprd 214 . . 3  |-  ( F  Fn  A  ->  (
( B  e.  A  /\  C  e.  B
)  ->  ( B  e.  dom  F  /\  C  e.  B ) ) )
5 smoel 6377 . . . 4  |-  ( ( Smo  F  /\  B  e.  dom  F  /\  C  e.  B )  ->  ( F `  C )  e.  ( F `  B
) )
653expib 1154 . . 3  |-  ( Smo 
F  ->  ( ( B  e.  dom  F  /\  C  e.  B )  ->  ( F `  C
)  e.  ( F `
 B ) ) )
74, 6sylan9 638 . 2  |-  ( ( F  Fn  A  /\  Smo  F )  ->  (
( B  e.  A  /\  C  e.  B
)  ->  ( F `  C )  e.  ( F `  B ) ) )
87imp 418 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 358    e. wcel 1684   dom cdm 4689    Fn wfn 5250   ` cfv 5255   Smo wsmo 6362
This theorem is referenced by:  smo11  6381  smoord  6382  smogt  6384  cofsmo  7895
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-tr 4114  df-ord 4395  df-iota 5219  df-fn 5258  df-fv 5263  df-smo 6363
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