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Definition df-isom 6337
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:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (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-isom
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 6329 . 2 wff 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)
71, 2, 5wf1o 6327 . . 3 wff 𝐻:𝐴1-1-onto𝐵
8 vx . . . . . . . 8 setvar 𝑥
98cv 1537 . . . . . . 7 class 𝑥
10 vy . . . . . . . 8 setvar 𝑦
1110cv 1537 . . . . . . 7 class 𝑦
129, 11, 3wbr 5033 . . . . . 6 wff 𝑥𝑅𝑦
139, 5cfv 6328 . . . . . . 7 class (𝐻𝑥)
1411, 5cfv 6328 . . . . . . 7 class (𝐻𝑦)
1513, 14, 4wbr 5033 . . . . . 6 wff (𝐻𝑥)𝑆(𝐻𝑦)
1612, 15wb 209 . . . . 5 wff (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))
1716, 10, 1wral 3109 . . . 4 wff 𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))
1817, 8, 1wral 3109 . . 3 wff 𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))
197, 18wa 399 . 2 wff (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
206, 19wb 209 1 wff (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
Colors of variables: wff setvar class
This definition is referenced by:  isoeq1  7053  isoeq2  7054  isoeq3  7055  isoeq4  7056  isoeq5  7057  nfiso  7058  isof1o  7059  isof1oidb  7060  isof1oopb  7061  isorel  7062  soisores  7063  soisoi  7064  isoid  7065  isocnv  7066  isocnv2  7067  isocnv3  7068  isores2  7069  isores3  7071  isotr  7072  isoini2  7075  f1oiso  7087  f1owe  7089  smoiso2  7993  alephiso  9513  compssiso  9789  negiso  11612  om2uzisoi  13321  icopnfhmeo  23551  reefiso  25046  logltb  25194  isoun  30464  xrmulc1cn  31281  wepwsolem  39973  alephiso2  40244  iso0  40998  fourierdlem54  42789  rrx2plordisom  45124
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