Definition df-isom 6551

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 6543	. 2 wff 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)
7	1, 2, 5	wf1o 6541	. . 3 wff 𝐻:𝐴–1-1-onto→𝐵
8		vx	. . . . . . . 8 setvar 𝑥
9	8	cv 1533	. . . . . . 7 class 𝑥
10		vy	. . . . . . . 8 setvar 𝑦
11	10	cv 1533	. . . . . . 7 class 𝑦
12	9, 11, 3	wbr 5142	. . . . . 6 wff 𝑥𝑅𝑦
13	9, 5	cfv 6542	. . . . . . 7 class (𝐻‘𝑥)
14	11, 5	cfv 6542	. . . . . . 7 class (𝐻‘𝑦)
15	13, 14, 4	wbr 5142	. . . . . 6 wff (𝐻‘𝑥)𝑆(𝐻‘𝑦)
16	12, 15	wb 205	. . . . 5 wff (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))
17	16, 10, 1	wral 3056	. . . 4 wff ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))
18	17, 8, 1	wral 3056	. . 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 205	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  7319  isoeq2  7320  isoeq3  7321  isoeq4  7322  isoeq5  7323  nfiso  7324  isof1o  7325  isof1oidb  7326  isof1oopb  7327  isorel  7328  soisores  7329  soisoi  7330  isoid  7331  isocnv  7332  isocnv2  7333  isocnv3  7334  isores2  7335  isores3  7337  isotr  7338  isoini2  7341  f1oiso  7353  f1owe  7355  smoiso2  8383  alephiso  10113  compssiso  10389  negiso  12216  om2uzisoi  13943  icopnfhmeo  24855  reefiso  26372  logltb  26521  om2noseqiso  28162  isoun  32465  mgcf1o  32712  xrmulc1cn  33467  sticksstones3  41552  wepwsolem  42388  alephiso2  42911  iso0  43667  fourierdlem54  45471  rrx2plordisom  47719