Theorem issmo2 6435

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 5485	. . . . 5 |- ( ( F : A --> B /\ B C_ On ) -> F : A --> On )
2	1	ex 115	. . . 4 |- ( F : A --> B -> ( B C_ On -> F : A --> On ) )
3		fdm 5479	. . . . 5 |- ( F : A --> B -> dom F = A )
4	3	feq2d 5461	. . . 4 |- ( F : A --> B -> ( F : dom F --> On <-> F : A --> On ) )
5	2, 4	sylibrd 169	. . 3 |- ( F : A --> B -> ( B C_ On -> F : dom F --> On ) )
6		ordeq 4463	. . . . 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 158	. . 3 |- ( F : A --> B -> ( Ord A -> Ord dom F ) )
9	3	raleqdv 2734	. . . 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 158	. . 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 1353	. 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 6433	. 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	imbitrrdi 162	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 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   Ord word 4453   Oncon0 4454   dom cdm 4719   -->wf 5314   ` cfv 5318   Smo wsmo 6431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-in 3203  df-ss 3210  df-uni 3889  df-tr 4183  df-iord 4457  df-fn 5321  df-f 5322  df-smo 6432

This theorem is referenced by: (None)