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Theorem smofvon 8306
Description: If 𝐡 is a strictly monotone ordinal function, and 𝐴 is in the domain of 𝐡, then the value of the function at 𝐴 is an ordinal. (Contributed by Andrew Salmon, 20-Nov-2011.)
Assertion
Ref Expression
smofvon ((Smo 𝐡 ∧ 𝐴 ∈ dom 𝐡) β†’ (π΅β€˜π΄) ∈ On)

Proof of Theorem smofvon
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-smo 8293 . . 3 (Smo 𝐡 ↔ (𝐡:dom 𝐡⟢On ∧ Ord dom 𝐡 ∧ βˆ€π‘₯ ∈ dom π΅βˆ€π‘¦ ∈ dom 𝐡(π‘₯ ∈ 𝑦 β†’ (π΅β€˜π‘₯) ∈ (π΅β€˜π‘¦))))
21simp1bi 1146 . 2 (Smo 𝐡 β†’ 𝐡:dom 𝐡⟢On)
32ffvelcdmda 7036 1 ((Smo 𝐡 ∧ 𝐴 ∈ dom 𝐡) β†’ (π΅β€˜π΄) ∈ On)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∈ wcel 2107  βˆ€wral 3061  dom cdm 5634  Ord word 6317  Oncon0 6318  βŸΆwf 6493  β€˜cfv 6497  Smo wsmo 8292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-smo 8293
This theorem is referenced by:  smoiun  8308
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