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Theorem smofvon 6392
Description: If  B is a strictly monotone ordinal function, and  A is in the domain of  B, then the value of the function at 
A is an ordinal. (Contributed by Andrew Salmon, 20-Nov-2011.)
Assertion
Ref Expression
smofvon  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( B `  A
)  e.  On )

Proof of Theorem smofvon
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-smo 6379 . . 3  |-  ( Smo 
B  <->  ( B : dom  B --> On  /\  Ord  dom 
B  /\  A. x  e.  dom  B A. y  e.  dom  B ( x  e.  y  ->  ( B `  x )  e.  ( B `  y
) ) ) )
21simp1bi 970 . 2  |-  ( Smo 
B  ->  B : dom  B --> On )
3 ffvelrn 5679 . 2  |-  ( ( B : dom  B --> On  /\  A  e.  dom  B )  ->  ( B `  A )  e.  On )
42, 3sylan 457 1  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( B `  A
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   A.wral 2556   Ord word 4407   Oncon0 4408   dom cdm 4705   -->wf 5267   ` cfv 5271   Smo wsmo 6378
This theorem is referenced by:  smoiun  6394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-smo 6379
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