Definition df-isom 6523

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 6515	. 2 wff 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)
7	1, 2, 5	wf1o 6513	. . 3 wff 𝐻:𝐴–1-1-onto→𝐵
8		vx	. . . . . . . 8 setvar 𝑥
9	8	cv 1539	. . . . . . 7 class 𝑥
10		vy	. . . . . . . 8 setvar 𝑦
11	10	cv 1539	. . . . . . 7 class 𝑦
12	9, 11, 3	wbr 5110	. . . . . 6 wff 𝑥𝑅𝑦
13	9, 5	cfv 6514	. . . . . . 7 class (𝐻‘𝑥)
14	11, 5	cfv 6514	. . . . . . 7 class (𝐻‘𝑦)
15	13, 14, 4	wbr 5110	. . . . . 6 wff (𝐻‘𝑥)𝑆(𝐻‘𝑦)
16	12, 15	wb 206	. . . . 5 wff (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))
17	16, 10, 1	wral 3045	. . . 4 wff ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))
18	17, 8, 1	wral 3045	. . 3 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))
19	7, 18	wa 395	. 2 wff (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
20	6, 19	wb 206	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 referenced by:  isoeq1  7295  isoeq2  7296  isoeq3  7297  isoeq4  7298  isoeq5  7299  nfiso  7300  isof1o  7301  isof1oidb  7302  isof1oopb  7303  isorel  7304  soisores  7305  soisoi  7306  isoid  7307  isocnv  7308  isocnv2  7309  isocnv3  7310  isores2  7311  isores3  7313  isotr  7314  isoini2  7317  f1oiso  7329  f1owe  7331  smoiso2  8341  alephiso  10058  compssiso  10334  negiso  12170  om2uzisoi  13926  icopnfhmeo  24848  reefiso  26365  logltb  26516  onsiso  28176  om2noseqiso  28203  isoun  32632  mgcf1o  32936  xrmulc1cn  33927  sticksstones3  42143  wepwsolem  43038  alephiso2  43554  iso0  44303  fourierdlem54  46165  rrx2plordisom  48716