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Theorem smoel2 6617
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 5536 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
21eleq2d 2502 . . . . 5  |-  ( F  Fn  A  ->  ( B  e.  dom  F  <->  B  e.  A ) )
32anbi1d 686 . . . 4  |-  ( F  Fn  A  ->  (
( B  e.  dom  F  /\  C  e.  B
)  <->  ( B  e.  A  /\  C  e.  B ) ) )
43biimprd 215 . . 3  |-  ( F  Fn  A  ->  (
( B  e.  A  /\  C  e.  B
)  ->  ( B  e.  dom  F  /\  C  e.  B ) ) )
5 smoel 6614 . . . 4  |-  ( ( Smo  F  /\  B  e.  dom  F  /\  C  e.  B )  ->  ( F `  C )  e.  ( F `  B
) )
653expib 1156 . . 3  |-  ( Smo 
F  ->  ( ( B  e.  dom  F  /\  C  e.  B )  ->  ( F `  C
)  e.  ( F `
 B ) ) )
74, 6sylan9 639 . 2  |-  ( ( F  Fn  A  /\  Smo  F )  ->  (
( B  e.  A  /\  C  e.  B
)  ->  ( F `  C )  e.  ( F `  B ) ) )
87imp 419 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 359    e. wcel 1725   dom cdm 4870    Fn wfn 5441   ` cfv 5446   Smo wsmo 6599
This theorem is referenced by:  smo11  6618  smoord  6619  smogt  6621  cofsmo  8141
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-tr 4295  df-ord 4576  df-iota 5410  df-fn 5449  df-fv 5454  df-smo 6600
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