Definition df-isom 5610

Description: Define the isomorphism predicate. We read this as " H is an R, S isomorphism of A onto B." Normally, R and S are ordering relations on A and B respectively. Definition 6.28 of [TakeutiZaring] p. 32, whose notation is the same as ours except that R and S are subscripts. (Contributed by NM, 4-Mar-1997.)

Assertion

Ref	Expression
df-isom	|- ( H Isom R , S ( A , B ) <-> ( H : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) ) )

Distinct variable groups:   x, y, A   x, B, y   x, R, y   x, S, y   x, H, y

Detailed syntax breakdown of Definition df-isom

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2		cB	. . 3 class B
3		cR	. . 3 class R
4		cS	. . 3 class S
5		cH	. . 3 class H
6	1, 2, 3, 4, 5	wiso 5602	. 2 wff H Isom R , S ( A , B )
7	1, 2, 5	wf1o 5600	. . 3 wff H : A -1-1-onto-> B
8		vx	. . . . . . . 8 setvar x
9	8	cv 1454	. . . . . . 7 class x
10		vy	. . . . . . . 8 setvar y
11	10	cv 1454	. . . . . . 7 class y
12	9, 11, 3	wbr 4416	. . . . . 6 wff x R y
13	9, 5	cfv 5601	. . . . . . 7 class ( H ` x )
14	11, 5	cfv 5601	. . . . . . 7 class ( H ` y )
15	13, 14, 4	wbr 4416	. . . . . 6 wff ( H ` x ) S ( H ` y )
16	12, 15	wb 189	. . . . 5 wff ( x R y <-> ( H ` x ) S ( H ` y ) )
17	16, 10, 1	wral 2749	. . . 4 wff A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) )
18	17, 8, 1	wral 2749	. . 3 wff A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) )
19	7, 18	wa 375	. 2 wff ( H : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) )
20	6, 19	wb 189	1 wff ( H Isom R , S ( A , B ) <-> ( H : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) ) )

This definition is referenced by:  isoeq1  6235  isoeq2  6236  isoeq3  6237  isoeq4  6238  isoeq5  6239  nfiso  6240  isof1o  6241  isorel  6242  soisores  6243  soisoi  6244  isoid  6245  isocnv  6246  isocnv2  6247  isocnv3  6248  isores2  6249  isores3  6251  isotr  6252  isoini2  6255  f1oiso  6267  f1owe  6269  smoiso2  7114  alephiso  8555  compssiso  8830  negiso  10615  om2uzisoi  12200  icopnfhmeo  22020  reefiso  23452  logltb  23598  isoun  28331  xrmulc1cn  28785  wepwsolem  35945  iso0  36699  fourierdlem54  38062