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Theorem smofvon 7739
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 𝐵𝐴 ∈ dom 𝐵) → (𝐵𝐴) ∈ On)

Proof of Theorem smofvon
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-smo 7726 . . 3 (Smo 𝐵 ↔ (𝐵:dom 𝐵⟶On ∧ Ord dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐵𝑦 ∈ dom 𝐵(𝑥𝑦 → (𝐵𝑥) ∈ (𝐵𝑦))))
21simp1bi 1136 . 2 (Smo 𝐵𝐵:dom 𝐵⟶On)
32ffvelrnda 6623 1 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝐵𝐴) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wcel 2107  wral 3090  dom cdm 5355  Ord word 5975  Oncon0 5976  wf 6131  cfv 6135  Smo wsmo 7725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-smo 7726
This theorem is referenced by:  smoiun  7741
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