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Theorem smofvon 6987
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 6974 . . 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 1012 . 2  |-  ( Smo 
B  ->  B : dom  B --> On )
32ffvelrnda 5965 1  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( B `  A
)  e.  On )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1842   A.wral 2753   Ord word 4820   Oncon0 4821   dom cdm 4942   -->wf 5521   ` cfv 5525   Smo wsmo 6973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-smo 6974
This theorem is referenced by:  smoiun  6989
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