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Definition df-isom 6552
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 6544 . 2 wff 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)
71, 2, 5wf1o 6542 . . 3 wff 𝐻:𝐴1-1-onto𝐵
8 vx . . . . . . . 8 setvar 𝑥
98cv 1539 . . . . . . 7 class 𝑥
10 vy . . . . . . . 8 setvar 𝑦
1110cv 1539 . . . . . . 7 class 𝑦
129, 11, 3wbr 5148 . . . . . 6 wff 𝑥𝑅𝑦
139, 5cfv 6543 . . . . . . 7 class (𝐻𝑥)
1411, 5cfv 6543 . . . . . . 7 class (𝐻𝑦)
1513, 14, 4wbr 5148 . . . . . 6 wff (𝐻𝑥)𝑆(𝐻𝑦)
1612, 15wb 205 . . . . 5 wff (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))
1716, 10, 1wral 3060 . . . 4 wff 𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))
1817, 8, 1wral 3060 . . 3 wff 𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))
197, 18wa 395 . 2 wff (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
206, 19wb 205 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  7317  isoeq2  7318  isoeq3  7319  isoeq4  7320  isoeq5  7321  nfiso  7322  isof1o  7323  isof1oidb  7324  isof1oopb  7325  isorel  7326  soisores  7327  soisoi  7328  isoid  7329  isocnv  7330  isocnv2  7331  isocnv3  7332  isores2  7333  isores3  7335  isotr  7336  isoini2  7339  f1oiso  7351  f1owe  7353  smoiso2  8375  alephiso  10099  compssiso  10375  negiso  12201  om2uzisoi  13926  icopnfhmeo  24788  reefiso  26300  logltb  26448  isoun  32357  mgcf1o  32607  xrmulc1cn  33375  sticksstones3  41433  wepwsolem  42249  alephiso2  42774  iso0  43531  fourierdlem54  45337  rrx2plordisom  47573
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