Theorem issmo2 6194

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 5292	. . . . 5 |- ( ( F : A --> B /\ B C_ On ) -> F : A --> On )
2	1	ex 114	. . . 4 |- ( F : A --> B -> ( B C_ On -> F : A --> On ) )
3		fdm 5286	. . . . 5 |- ( F : A --> B -> dom F = A )
4	3	feq2d 5268	. . . 4 |- ( F : A --> B -> ( F : dom F --> On <-> F : A --> On ) )
5	2, 4	sylibrd 168	. . 3 |- ( F : A --> B -> ( B C_ On -> F : dom F --> On ) )
6		ordeq 4302	. . . . 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 157	. . 3 |- ( F : A --> B -> ( Ord A -> Ord dom F ) )
9	3	raleqdv 2635	. . . 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 157	. . 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 1298	. 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 6192	. 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 161	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 104   /\ w3a 963   = wceq 1332   e. wcel 1481   A.wral 2417   C_ wss 3076   Ord word 4292   Oncon0 4293   dom cdm 4547   -->wf 5127   ` cfv 5131   Smo wsmo 6190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3082  df-ss 3089  df-uni 3745  df-tr 4035  df-iord 4296  df-fn 5134  df-f 5135  df-smo 6191

This theorem is referenced by: (None)