Definition df-isom 6511

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 6503	. 2 wff 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)
7	1, 2, 5	wf1o 6501	. . 3 wff 𝐻:𝐴–1-1-onto→𝐵
8		vx	. . . . . . . 8 setvar 𝑥
9	8	cv 1541	. . . . . . 7 class 𝑥
10		vy	. . . . . . . 8 setvar 𝑦
11	10	cv 1541	. . . . . . 7 class 𝑦
12	9, 11, 3	wbr 5100	. . . . . 6 wff 𝑥𝑅𝑦
13	9, 5	cfv 6502	. . . . . . 7 class (𝐻‘𝑥)
14	11, 5	cfv 6502	. . . . . . 7 class (𝐻‘𝑦)
15	13, 14, 4	wbr 5100	. . . . . 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 3052	. . . 4 wff ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))
18	17, 8, 1	wral 3052	. . 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  7275  isoeq2  7276  isoeq3  7277  isoeq4  7278  isoeq5  7279  nfiso  7280  isof1o  7281  isof1oidb  7282  isof1oopb  7283  isorel  7284  soisores  7285  soisoi  7286  isoid  7287  isocnv  7288  isocnv2  7289  isocnv3  7290  isores2  7291  isores3  7293  isotr  7294  isoini2  7297  f1oiso  7309  f1owe  7311  smoiso2  8313  alephiso  10022  compssiso  10298  negiso  12136  om2uzisoi  13891  icopnfhmeo  24914  reefiso  26431  logltb  26582  oniso  28284  om2noseqiso  28315  isoun  32798  mgcf1o  33102  xrmulc1cn  34114  sticksstones3  42547  wepwsolem  43428  alephiso2  43943  iso0  44692  fourierdlem54  46547  rrx2plordisom  49112