Definition df-smo 6374

Description: Definition of a strictly monotone ordinal function. Definition 7.46 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 15-Nov-2011.)

Assertion

Ref	Expression
df-smo	|- ( Smo A <-> ( A : dom A --> On /\ Ord dom A /\ A. x e. dom A A. y e. dom A ( x e. y -> ( A ` x ) e. ( A ` y ) ) ) )

Distinct variable group:   x, y, A

Detailed syntax breakdown of Definition df-smo

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2	1	wsmo 6373	. 2 wff Smo A
3	1	cdm 4676	. . . 4 class dom A
4		con0 4411	. . . 4 class On
5	3, 4, 1	wf 5268	. . 3 wff A : dom A --> On
6	3	word 4410	. . 3 wff Ord dom A
7		vx	. . . . . . 7 setvar x
8		vy	. . . . . . 7 setvar y
9	7, 8	wel 2177	. . . . . 6 wff x e. y
10	7	cv 1372	. . . . . . . 8 class x
11	10, 1	cfv 5272	. . . . . . 7 class ( A ` x )
12	8	cv 1372	. . . . . . . 8 class y
13	12, 1	cfv 5272	. . . . . . 7 class ( A ` y )
14	11, 13	wcel 2176	. . . . . 6 wff ( A ` x ) e. ( A ` y )
15	9, 14	wi 4	. . . . 5 wff ( x e. y -> ( A ` x ) e. ( A ` y ) )
16	15, 8, 3	wral 2484	. . . 4 wff A. y e. dom A ( x e. y -> ( A ` x ) e. ( A ` y ) )
17	16, 7, 3	wral 2484	. . 3 wff A. x e. dom A A. y e. dom A ( x e. y -> ( A ` x ) e. ( A ` y ) )
18	5, 6, 17	w3a 981	. 2 wff ( A : dom A --> On /\ Ord dom A /\ A. x e. dom A A. y e. dom A ( x e. y -> ( A ` x ) e. ( A ` y ) ) )
19	2, 18	wb 105	1 wff ( Smo A <-> ( A : dom A --> On /\ Ord dom A /\ A. x e. dom A A. y e. dom A ( x e. y -> ( A ` x ) e. ( A ` y ) ) ) )

This definition is referenced by:  dfsmo2  6375  issmo  6376  smoeq  6378  smodm  6379  smores  6380  smofvon  6387  smoel  6388  smoiso  6390