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Theorem issmo2 5845
Description: Alternative definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.)
Assertion
Ref Expression
issmo2 (𝐹:AB → ((B ⊆ On Ord A x A y x (𝐹y) (𝐹x)) → Smo 𝐹))
Distinct variable groups:   x,A   x,𝐹,y
Allowed substitution hints:   A(y)   B(x,y)

Proof of Theorem issmo2
StepHypRef Expression
1 fss 4997 . . . . 5 ((𝐹:AB B ⊆ On) → 𝐹:A⟶On)
21ex 108 . . . 4 (𝐹:AB → (B ⊆ On → 𝐹:A⟶On))
3 fdm 4993 . . . . 5 (𝐹:AB → dom 𝐹 = A)
43feq2d 4978 . . . 4 (𝐹:AB → (𝐹:dom 𝐹⟶On ↔ 𝐹:A⟶On))
52, 4sylibrd 158 . . 3 (𝐹:AB → (B ⊆ On → 𝐹:dom 𝐹⟶On))
6 ordeq 4075 . . . . 5 (dom 𝐹 = A → (Ord dom 𝐹 ↔ Ord A))
73, 6syl 14 . . . 4 (𝐹:AB → (Ord dom 𝐹 ↔ Ord A))
87biimprd 147 . . 3 (𝐹:AB → (Ord A → Ord dom 𝐹))
93raleqdv 2505 . . . 4 (𝐹:AB → (x dom 𝐹y x (𝐹y) (𝐹x) ↔ x A y x (𝐹y) (𝐹x)))
109biimprd 147 . . 3 (𝐹:AB → (x A y x (𝐹y) (𝐹x) → x dom 𝐹y x (𝐹y) (𝐹x)))
115, 8, 103anim123d 1213 . 2 (𝐹:AB → ((B ⊆ On Ord A x A y x (𝐹y) (𝐹x)) → (𝐹:dom 𝐹⟶On Ord dom 𝐹 x dom 𝐹y x (𝐹y) (𝐹x))))
12 dfsmo2 5843 . 2 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On Ord dom 𝐹 x dom 𝐹y x (𝐹y) (𝐹x)))
1311, 12syl6ibr 151 1 (𝐹:AB → ((B ⊆ On Ord A x A y x (𝐹y) (𝐹x)) → Smo 𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   w3a 884   = wceq 1242   wcel 1390  wral 2300  wss 2911  Ord word 4065  Oncon0 4066  dom cdm 4288  wf 4841  cfv 4845  Smo wsmo 5841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846  df-iord 4069  df-fn 4848  df-f 4849  df-smo 5842
This theorem is referenced by: (None)
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