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Mirrors > Home > MPE Home > Th. List > df-isom | Structured version Visualization version GIF version |
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.) |
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)))) |
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 6350 | . 2 wff 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) |
7 | 1, 2, 5 | wf1o 6348 | . . 3 wff 𝐻:𝐴–1-1-onto→𝐵 |
8 | vx | . . . . . . . 8 setvar 𝑥 | |
9 | 8 | cv 1527 | . . . . . . 7 class 𝑥 |
10 | vy | . . . . . . . 8 setvar 𝑦 | |
11 | 10 | cv 1527 | . . . . . . 7 class 𝑦 |
12 | 9, 11, 3 | wbr 5058 | . . . . . 6 wff 𝑥𝑅𝑦 |
13 | 9, 5 | cfv 6349 | . . . . . . 7 class (𝐻‘𝑥) |
14 | 11, 5 | cfv 6349 | . . . . . . 7 class (𝐻‘𝑦) |
15 | 13, 14, 4 | wbr 5058 | . . . . . 6 wff (𝐻‘𝑥)𝑆(𝐻‘𝑦) |
16 | 12, 15 | wb 207 | . . . . 5 wff (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) |
17 | 16, 10, 1 | wral 3138 | . . . 4 wff ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) |
18 | 17, 8, 1 | wral 3138 | . . 3 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) |
19 | 7, 18 | wa 396 | . 2 wff (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) |
20 | 6, 19 | wb 207 | 1 wff (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) |
Colors of variables: wff setvar class |
This definition is referenced by: isoeq1 7059 isoeq2 7060 isoeq3 7061 isoeq4 7062 isoeq5 7063 nfiso 7064 isof1o 7065 isof1oidb 7066 isof1oopb 7067 isorel 7068 soisores 7069 soisoi 7070 isoid 7071 isocnv 7072 isocnv2 7073 isocnv3 7074 isores2 7075 isores3 7077 isotr 7078 isoini2 7081 f1oiso 7093 f1owe 7095 smoiso2 7997 alephiso 9513 compssiso 9785 negiso 11610 om2uzisoi 13312 icopnfhmeo 23476 reefiso 24965 logltb 25110 isoun 30364 xrmulc1cn 31073 wepwsolem 39522 alephiso2 39797 iso0 40519 fourierdlem54 42326 rrx2plordisom 44608 |
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