ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  issmo2 Structured version   GIF version

Theorem issmo2 5814
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 4968 . . . . 5 ((𝐹:AB B ⊆ On) → 𝐹:A⟶On)
21ex 108 . . . 4 (𝐹:AB → (B ⊆ On → 𝐹:A⟶On))
3 fdm 4964 . . . . 5 (𝐹:AB → dom 𝐹 = A)
43feq2d 4949 . . . 4 (𝐹:AB → (𝐹:dom 𝐹⟶On ↔ 𝐹:A⟶On))
52, 4sylibrd 158 . . 3 (𝐹:AB → (B ⊆ On → 𝐹:dom 𝐹⟶On))
6 ordeq 4047 . . . . 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 2480 . . . 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 1194 . 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 5812 . 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 867   = wceq 1223   wcel 1366  wral 2275  wss 2885  Ord word 4037  Oncon0 4038  dom cdm 4260  wf 4813  cfv 4817  Smo wsmo 5810
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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-in 2892  df-ss 2899  df-uni 3544  df-tr 3818  df-iord 4041  df-fn 4820  df-f 4821  df-smo 5811
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator