Theorem issmo2 5927

Description: Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.)

Assertion

Ref	Expression
issmo2	|- ( F : A --> B -> ( ( B C_ On /\ Ord A /\ A. x e. A A. y e. x ( F ` y ) e. ( F ` x ) ) -> Smo F ) )

Distinct variable groups:   x, A   x, F, y

Allowed substitution hints:   A( y)   B( x, y)

Proof of Theorem issmo2

Step	Hyp	Ref	Expression
1		fss 5074	. . . . 5 |- ( ( F : A --> B /\ B C_ On ) -> F : A --> On )
2	1	ex 113	. . . 4 |- ( F : A --> B -> ( B C_ On -> F : A --> On ) )
3		fdm 5070	. . . . 5 |- ( F : A --> B -> dom F = A )
4	3	feq2d 5055	. . . 4 |- ( F : A --> B -> ( F : dom F --> On <-> F : A --> On ) )
5	2, 4	sylibrd 167	. . 3 |- ( F : A --> B -> ( B C_ On -> F : dom F --> On ) )
6		ordeq 4127	. . . . 5 |- ( dom F = A -> ( Ord dom F <-> Ord A ) )
7	3, 6	syl 14	. . . 4 |- ( F : A --> B -> ( Ord dom F <-> Ord A ) )
8	7	biimprd 156	. . 3 |- ( F : A --> B -> ( Ord A -> Ord dom F ) )
9	3	raleqdv 2555	. . . 4 |- ( F : A --> B -> ( A. x e. dom F A. y e. x ( F ` y ) e. ( F ` x ) <-> A. x e. A A. y e. x ( F ` y ) e. ( F ` x ) ) )
10	9	biimprd 156	. . 3 |- ( F : A --> B -> ( A. x e. A A. y e. x ( F ` y ) e. ( F ` x ) -> A. x e. dom F A. y e. x ( F ` y ) e. ( F ` x ) ) )
11	5, 8, 10	3anim123d 1250	. 2 |- ( F : A --> B -> ( ( B C_ On /\ Ord A /\ A. x e. A A. y e. x ( F ` y ) e. ( F ` x ) ) -> ( F : dom F --> On /\ Ord dom F /\ A. x e. dom F A. y e. x ( F ` y ) e. ( F ` x ) ) ) )
12		dfsmo2 5925	. 2 |- ( Smo F <-> ( F : dom F --> On /\ Ord dom F /\ A. x e. dom F A. y e. x ( F ` y ) e. ( F ` x ) ) )
13	11, 12	syl6ibr 160	1 |- ( F : A --> B -> ( ( B C_ On /\ Ord A /\ A. x e. A A. y e. x ( F ` y ) e. ( F ` x ) ) -> Smo F ) )

Syntax hints:   -> wi 4   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   A.wral 2348   C_ wss 2973   Ord word 4117   Oncon0 4118   dom cdm 4363   -->wf 4918   ` cfv 4922   Smo wsmo 5923

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602  df-tr 3876  df-iord 4121  df-fn 4925  df-f 4926  df-smo 5924

This theorem is referenced by: (None)