Theorem issmo2 6398

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 5457	. . . . 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 5451	. . . . 5 |- ( F : A --> B -> dom F = A )
4	3	feq2d 5433	. . . 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 4437	. . . . 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 2711	. . . 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 1332	. 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 6396	. 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 981   = wceq 1373   e. wcel 2178   A.wral 2486   C_ wss 3174   Ord word 4427   Oncon0 4428   dom cdm 4693   -->wf 5286   ` cfv 5290   Smo wsmo 6394

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-in 3180  df-ss 3187  df-uni 3865  df-tr 4159  df-iord 4431  df-fn 5293  df-f 5294  df-smo 6395

This theorem is referenced by: (None)