MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  smofvon Structured version   Visualization version   GIF version

Theorem smofvon 8358
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 8345 . . 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 1145 . 2 (Smo 𝐡 β†’ 𝐡:dom 𝐡⟢On)
32ffvelcdmda 7086 1 ((Smo 𝐡 ∧ 𝐴 ∈ dom 𝐡) β†’ (π΅β€˜π΄) ∈ On)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∈ wcel 2106  βˆ€wral 3061  dom cdm 5676  Ord word 6363  Oncon0 6364  βŸΆwf 6539  β€˜cfv 6543  Smo wsmo 8344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-smo 8345
This theorem is referenced by:  smoiun  8360
  Copyright terms: Public domain W3C validator