MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  smofvon2 Structured version   Visualization version   GIF version

Theorem smofvon2 8303
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 8294 . . . 4 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟢On ∧ Ord dom 𝐹 ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘¦ ∈ π‘₯ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)))
21simp1bi 1146 . . 3 (Smo 𝐹 β†’ 𝐹:dom 𝐹⟢On)
3 ffvelcdm 7033 . . . 4 ((𝐹:dom 𝐹⟢On ∧ 𝐡 ∈ dom 𝐹) β†’ (πΉβ€˜π΅) ∈ On)
43expcom 415 . . 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 6878 . . . 4 (Β¬ 𝐡 ∈ dom 𝐹 β†’ (πΉβ€˜π΅) = βˆ…)
7 0elon 6372 . . . 4 βˆ… ∈ On
86, 7eqeltrdi 2842 . . 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 2107  βˆ€wral 3061  βˆ…c0 4283  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-11 2155  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-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-tr 5224  df-id 5532  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-ord 6321  df-on 6322  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-smo 8293
This theorem is referenced by:  smo11  8311  smoord  8312  smoword  8313  smogt  8314
  Copyright terms: Public domain W3C validator