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Theorem smofvon 8386
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 8373 . . 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 1142 . 2 (Smo 𝐡 β†’ 𝐡:dom 𝐡⟢On)
32ffvelcdmda 7099 1 ((Smo 𝐡 ∧ 𝐴 ∈ dom 𝐡) β†’ (π΅β€˜π΄) ∈ On)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∈ wcel 2098  βˆ€wral 3058  dom cdm 5682  Ord word 6373  Oncon0 6374  βŸΆwf 6549  β€˜cfv 6553  Smo wsmo 8372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-smo 8373
This theorem is referenced by:  smoiun  8388
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