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Theorem smofvon 7096
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 7083 . . 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 1045 . 2  |-  ( Smo 
B  ->  B : dom  B --> On )
32ffvelrnda 6037 1  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( B `  A
)  e.  On )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    e. wcel 1904   A.wral 2756   dom cdm 4839   Ord word 5429   Oncon0 5430   -->wf 5585   ` cfv 5589   Smo wsmo 7082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-smo 7083
This theorem is referenced by:  smoiun  7098
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