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Theorem smofvon2 8358
Description: The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)
Assertion
Ref Expression
smofvon2 (Smo 𝐹 β†’ (πΉβ€˜π΅) ∈ On)
Proof of Theorem smofvon2
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsmo2 8349 . . . 4 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟢On ∧ Ord dom 𝐹 ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘¦ ∈ π‘₯ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)))
21simp1bi 1145 . . 3 (Smo 𝐹 β†’ 𝐹:dom 𝐹⟢On)
3 ffvelcdm 7083 . . . 4 ((𝐹:dom 𝐹⟢On ∧ 𝐡 ∈ dom 𝐹) β†’ (πΉβ€˜π΅) ∈ On)
43expcom 414 . . 3 (𝐡 ∈ dom 𝐹 β†’ (𝐹:dom 𝐹⟢On β†’ (πΉβ€˜π΅) ∈ On))
52, 4syl5 34 . 2 (𝐡 ∈ dom 𝐹 β†’ (Smo 𝐹 β†’ (πΉβ€˜π΅) ∈ On))
6 ndmfv 6926 . . . 4 (Β¬ 𝐡 ∈ dom 𝐹 β†’ (πΉβ€˜π΅) = βˆ…)
7 0elon 6418 . . . 4 βˆ… ∈ On
86, 7eqeltrdi 2841 . . 3 (Β¬ 𝐡 ∈ dom 𝐹 β†’ (πΉβ€˜π΅) ∈ On)
98a1d 25 . 2 (Β¬ 𝐡 ∈ dom 𝐹 β†’ (Smo 𝐹 β†’ (πΉβ€˜π΅) ∈ On))
105, 9pm2.61i 182 1 (Smo 𝐹 β†’ (πΉβ€˜π΅) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∈ wcel 2106  βˆ€wral 3061  βˆ…c0 4322  dom cdm 5676  Ord word 6363  Oncon0 6364  βŸΆwf 6539  β€˜cfv 6543  Smo wsmo 8347
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-11 2154  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-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-id 5574  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-ord 6367  df-on 6368  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-smo 8348
This theorem is referenced by:  smo11  8366  smoord  8367  smoword  8368  smogt  8369
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