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Definition df-iso 4796
 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 SF, 5-Jan-2015.)
Assertion
Ref Expression
df-iso (H Isom R, S (A, B) ↔ (H:A1-1-ontoB x A y A (xRy ↔ (Hx)S(Hy))))
Distinct variable groups:   x,y,A   x,B,y   x,R,y   x,S,y   x,H,y

Detailed syntax breakdown of Definition df-iso
StepHypRef 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
61, 2, 3, 4, 5wiso 4782 . 2 wff H Isom R, S (A, B)
71, 2, 5wf1o 4780 . . 3 wff H:A1-1-ontoB
8 vx . . . . . . . 8 setvar x
98cv 1641 . . . . . . 7 class x
10 vy . . . . . . . 8 setvar y
1110cv 1641 . . . . . . 7 class y
129, 11, 3wbr 4639 . . . . . 6 wff xRy
139, 5cfv 4781 . . . . . . 7 class (Hx)
1411, 5cfv 4781 . . . . . . 7 class (Hy)
1513, 14, 4wbr 4639 . . . . . 6 wff (Hx)S(Hy)
1612, 15wb 176 . . . . 5 wff (xRy ↔ (Hx)S(Hy))
1716, 10, 1wral 2614 . . . 4 wff y A (xRy ↔ (Hx)S(Hy))
1817, 8, 1wral 2614 . . 3 wff x A y A (xRy ↔ (Hx)S(Hy))
197, 18wa 358 . 2 wff (H:A1-1-ontoB x A y A (xRy ↔ (Hx)S(Hy)))
206, 19wb 176 1 wff (H Isom R, S (A, B) ↔ (H:A1-1-ontoB x A y A (xRy ↔ (Hx)S(Hy))))
 Colors of variables: wff setvar class This definition is referenced by:  isoeq1  5482  isoeq2  5483  isoeq3  5484  isoeq4  5485  isoeq5  5486  nfiso  5487  isof1o  5488  isorel  5489  isoid  5490  isocnv  5491  isocnv2  5492  isores2  5493  isotr  5495  isoini2  5498  f1oiso  5499
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