Definition df-isom 5386

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 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (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 5378	. 2 wff 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)
7	1, 2, 5	wf1o 5376	. . 3 wff 𝐻:𝐴–1-1-onto→𝐵
8		vx	. . . . . . . 8 setvar 𝑥
9	8	cv 1401	. . . . . . 7 class 𝑥
10		vy	. . . . . . . 8 setvar 𝑦
11	10	cv 1401	. . . . . . 7 class 𝑦
12	9, 11, 3	wbr 4130	. . . . . 6 wff 𝑥𝑅𝑦
13	9, 5	cfv 5377	. . . . . . 7 class (𝐻‘𝑥)
14	11, 5	cfv 5377	. . . . . . 7 class (𝐻‘𝑦)
15	13, 14, 4	wbr 4130	. . . . . 6 wff (𝐻‘𝑥)𝑆(𝐻‘𝑦)
16	12, 15	wb 105	. . . . 5 wff (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))
17	16, 10, 1	wral 2528	. . . 4 wff ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))
18	17, 8, 1	wral 2528	. . 3 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))
19	7, 18	wa 104	. 2 wff (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
20	6, 19	wb 105	1 wff (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))

This definition is used by:  isoeq1  6007  isoeq2  6008  isoeq3  6009  isoeq4  6010  isoeq5  6011  nfiso  6012  isof1o  6013  isorel  6014  isoid  6016  isocnv  6017  isocnv2  6018  isores2  6019  isores3  6021  isotr  6022  iso0  6023  isoini2  6025  f1oiso  6032  negiso  9285  frec2uzisod  10844  zfz1isolem1  11292  xrnegiso  12028  reefiso  15878  logltb  15975