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Theorem dfsmo2 8351
Description: Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 4-Mar-2013.)
Assertion
Ref Expression
dfsmo2 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟢On ∧ Ord dom 𝐹 ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘¦ ∈ π‘₯ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)))
Distinct variable group:   π‘₯,𝐹,𝑦

Proof of Theorem dfsmo2
StepHypRef Expression
1 df-smo 8350 . 2 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟢On ∧ Ord dom 𝐹 ∧ βˆ€π‘¦ ∈ dom πΉβˆ€π‘₯ ∈ dom 𝐹(𝑦 ∈ π‘₯ β†’ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯))))
2 ralcom 3284 . . . . . 6 (βˆ€π‘¦ ∈ dom πΉβˆ€π‘₯ ∈ dom 𝐹(𝑦 ∈ π‘₯ β†’ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)) ↔ βˆ€π‘₯ ∈ dom πΉβˆ€π‘¦ ∈ dom 𝐹(𝑦 ∈ π‘₯ β†’ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)))
3 impexp 449 . . . . . . . . 9 (((𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ π‘₯) β†’ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)) ↔ (𝑦 ∈ dom 𝐹 β†’ (𝑦 ∈ π‘₯ β†’ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯))))
4 simpr 483 . . . . . . . . . . 11 ((𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ π‘₯) β†’ 𝑦 ∈ π‘₯)
5 ordtr1 6408 . . . . . . . . . . . . . . 15 (Ord dom 𝐹 β†’ ((𝑦 ∈ π‘₯ ∧ π‘₯ ∈ dom 𝐹) β†’ 𝑦 ∈ dom 𝐹))
653impib 1114 . . . . . . . . . . . . . 14 ((Ord dom 𝐹 ∧ 𝑦 ∈ π‘₯ ∧ π‘₯ ∈ dom 𝐹) β†’ 𝑦 ∈ dom 𝐹)
763com23 1124 . . . . . . . . . . . . 13 ((Ord dom 𝐹 ∧ π‘₯ ∈ dom 𝐹 ∧ 𝑦 ∈ π‘₯) β†’ 𝑦 ∈ dom 𝐹)
8 simp3 1136 . . . . . . . . . . . . 13 ((Ord dom 𝐹 ∧ π‘₯ ∈ dom 𝐹 ∧ 𝑦 ∈ π‘₯) β†’ 𝑦 ∈ π‘₯)
97, 8jca 510 . . . . . . . . . . . 12 ((Ord dom 𝐹 ∧ π‘₯ ∈ dom 𝐹 ∧ 𝑦 ∈ π‘₯) β†’ (𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ π‘₯))
1093expia 1119 . . . . . . . . . . 11 ((Ord dom 𝐹 ∧ π‘₯ ∈ dom 𝐹) β†’ (𝑦 ∈ π‘₯ β†’ (𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ π‘₯)))
114, 10impbid2 225 . . . . . . . . . 10 ((Ord dom 𝐹 ∧ π‘₯ ∈ dom 𝐹) β†’ ((𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ π‘₯) ↔ 𝑦 ∈ π‘₯))
1211imbi1d 340 . . . . . . . . 9 ((Ord dom 𝐹 ∧ π‘₯ ∈ dom 𝐹) β†’ (((𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ π‘₯) β†’ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)) ↔ (𝑦 ∈ π‘₯ β†’ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯))))
133, 12bitr3id 284 . . . . . . . 8 ((Ord dom 𝐹 ∧ π‘₯ ∈ dom 𝐹) β†’ ((𝑦 ∈ dom 𝐹 β†’ (𝑦 ∈ π‘₯ β†’ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯))) ↔ (𝑦 ∈ π‘₯ β†’ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯))))
1413ralbidv2 3171 . . . . . . 7 ((Ord dom 𝐹 ∧ π‘₯ ∈ dom 𝐹) β†’ (βˆ€π‘¦ ∈ dom 𝐹(𝑦 ∈ π‘₯ β†’ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)) ↔ βˆ€π‘¦ ∈ π‘₯ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)))
1514ralbidva 3173 . . . . . 6 (Ord dom 𝐹 β†’ (βˆ€π‘₯ ∈ dom πΉβˆ€π‘¦ ∈ dom 𝐹(𝑦 ∈ π‘₯ β†’ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)) ↔ βˆ€π‘₯ ∈ dom πΉβˆ€π‘¦ ∈ π‘₯ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)))
162, 15bitrid 282 . . . . 5 (Ord dom 𝐹 β†’ (βˆ€π‘¦ ∈ dom πΉβˆ€π‘₯ ∈ dom 𝐹(𝑦 ∈ π‘₯ β†’ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)) ↔ βˆ€π‘₯ ∈ dom πΉβˆ€π‘¦ ∈ π‘₯ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)))
1716pm5.32i 573 . . . 4 ((Ord dom 𝐹 ∧ βˆ€π‘¦ ∈ dom πΉβˆ€π‘₯ ∈ dom 𝐹(𝑦 ∈ π‘₯ β†’ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯))) ↔ (Ord dom 𝐹 ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘¦ ∈ π‘₯ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)))
1817anbi2i 621 . . 3 ((𝐹:dom 𝐹⟢On ∧ (Ord dom 𝐹 ∧ βˆ€π‘¦ ∈ dom πΉβˆ€π‘₯ ∈ dom 𝐹(𝑦 ∈ π‘₯ β†’ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)))) ↔ (𝐹:dom 𝐹⟢On ∧ (Ord dom 𝐹 ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘¦ ∈ π‘₯ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯))))
19 3anass 1093 . . 3 ((𝐹:dom 𝐹⟢On ∧ Ord dom 𝐹 ∧ βˆ€π‘¦ ∈ dom πΉβˆ€π‘₯ ∈ dom 𝐹(𝑦 ∈ π‘₯ β†’ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯))) ↔ (𝐹:dom 𝐹⟢On ∧ (Ord dom 𝐹 ∧ βˆ€π‘¦ ∈ dom πΉβˆ€π‘₯ ∈ dom 𝐹(𝑦 ∈ π‘₯ β†’ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)))))
20 3anass 1093 . . 3 ((𝐹:dom 𝐹⟢On ∧ Ord dom 𝐹 ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘¦ ∈ π‘₯ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)) ↔ (𝐹:dom 𝐹⟢On ∧ (Ord dom 𝐹 ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘¦ ∈ π‘₯ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯))))
2118, 19, 203bitr4i 302 . 2 ((𝐹:dom 𝐹⟢On ∧ Ord dom 𝐹 ∧ βˆ€π‘¦ ∈ dom πΉβˆ€π‘₯ ∈ dom 𝐹(𝑦 ∈ π‘₯ β†’ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯))) ↔ (𝐹:dom 𝐹⟢On ∧ Ord dom 𝐹 ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘¦ ∈ π‘₯ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)))
221, 21bitri 274 1 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟢On ∧ Ord dom 𝐹 ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘¦ ∈ π‘₯ (πΉβ€˜π‘¦) ∈ (πΉβ€˜π‘₯)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   ∈ wcel 2104  βˆ€wral 3059  dom cdm 5677  Ord word 6364  Oncon0 6365  βŸΆwf 6540  β€˜cfv 6544  Smo wsmo 8349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-11 2152  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1087  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-v 3474  df-in 3956  df-ss 3966  df-uni 4910  df-tr 5267  df-ord 6368  df-smo 8350
This theorem is referenced by:  issmo2  8353  smores2  8358  smofvon2  8360
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