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Theorem issmo2 8296
Description: Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.)
Assertion
Ref Expression
issmo2 (𝐹:𝐴⟢𝐡 β†’ ((𝐡 βŠ† On ∧ Ord 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ π‘₯ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)) β†’ Smo 𝐹))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐹,𝑦
Allowed substitution hints:   𝐴(𝑦)   𝐡(π‘₯,𝑦)

Proof of Theorem issmo2
StepHypRef Expression
1 fss 6686 . . . . 5 ((𝐹:𝐴⟢𝐡 ∧ 𝐡 βŠ† On) β†’ 𝐹:𝐴⟢On)
21ex 414 . . . 4 (𝐹:𝐴⟢𝐡 β†’ (𝐡 βŠ† On β†’ 𝐹:𝐴⟢On))
3 fdm 6678 . . . . 5 (𝐹:𝐴⟢𝐡 β†’ dom 𝐹 = 𝐴)
43feq2d 6655 . . . 4 (𝐹:𝐴⟢𝐡 β†’ (𝐹:dom 𝐹⟢On ↔ 𝐹:𝐴⟢On))
52, 4sylibrd 259 . . 3 (𝐹:𝐴⟢𝐡 β†’ (𝐡 βŠ† On β†’ 𝐹:dom 𝐹⟢On))
6 ordeq 6325 . . . . 5 (dom 𝐹 = 𝐴 β†’ (Ord dom 𝐹 ↔ Ord 𝐴))
73, 6syl 17 . . . 4 (𝐹:𝐴⟢𝐡 β†’ (Ord dom 𝐹 ↔ Ord 𝐴))
87biimprd 248 . . 3 (𝐹:𝐴⟢𝐡 β†’ (Ord 𝐴 β†’ Ord dom 𝐹))
93raleqdv 3312 . . . 4 (𝐹:𝐴⟢𝐡 β†’ (βˆ€π‘₯ ∈ dom πΉβˆ€π‘¦ ∈ π‘₯ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯) ↔ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ π‘₯ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)))
109biimprd 248 . . 3 (𝐹:𝐴⟢𝐡 β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ π‘₯ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯) β†’ βˆ€π‘₯ ∈ dom πΉβˆ€π‘¦ ∈ π‘₯ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)))
115, 8, 103anim123d 1444 . 2 (𝐹:𝐴⟢𝐡 β†’ ((𝐡 βŠ† On ∧ Ord 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ π‘₯ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)) β†’ (𝐹:dom 𝐹⟢On ∧ Ord dom 𝐹 ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘¦ ∈ π‘₯ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯))))
12 dfsmo2 8294 . 2 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟢On ∧ Ord dom 𝐹 ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘¦ ∈ π‘₯ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)))
1311, 12syl6ibr 252 1 (𝐹:𝐴⟢𝐡 β†’ ((𝐡 βŠ† On ∧ Ord 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ π‘₯ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)) β†’ Smo 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   βŠ† wss 3911  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-11 2155  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-v 3446  df-in 3918  df-ss 3928  df-uni 4867  df-tr 5224  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-ord 6321  df-fn 6500  df-f 6501  df-smo 8293
This theorem is referenced by:  alephsmo  10043  cofsmo  10210  cfsmolem  10211
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