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Mirrors > Home > ILE Home > Th. List > df-isom | 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 5124 | . 2 wff 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) |
7 | 1, 2, 5 | wf1o 5122 | . . 3 wff 𝐻:𝐴–1-1-onto→𝐵 |
8 | vx | . . . . . . . 8 setvar 𝑥 | |
9 | 8 | cv 1330 | . . . . . . 7 class 𝑥 |
10 | vy | . . . . . . . 8 setvar 𝑦 | |
11 | 10 | cv 1330 | . . . . . . 7 class 𝑦 |
12 | 9, 11, 3 | wbr 3929 | . . . . . 6 wff 𝑥𝑅𝑦 |
13 | 9, 5 | cfv 5123 | . . . . . . 7 class (𝐻‘𝑥) |
14 | 11, 5 | cfv 5123 | . . . . . . 7 class (𝐻‘𝑦) |
15 | 13, 14, 4 | wbr 3929 | . . . . . 6 wff (𝐻‘𝑥)𝑆(𝐻‘𝑦) |
16 | 12, 15 | wb 104 | . . . . 5 wff (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) |
17 | 16, 10, 1 | wral 2416 | . . . 4 wff ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) |
18 | 17, 8, 1 | wral 2416 | . . 3 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) |
19 | 7, 18 | wa 103 | . 2 wff (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) |
20 | 6, 19 | wb 104 | 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 set class |
This definition is referenced by: isoeq1 5702 isoeq2 5703 isoeq3 5704 isoeq4 5705 isoeq5 5706 nfiso 5707 isof1o 5708 isorel 5709 isoid 5711 isocnv 5712 isocnv2 5713 isores2 5714 isores3 5716 isotr 5717 iso0 5718 isoini2 5720 f1oiso 5727 negiso 8713 frec2uzisod 10180 zfz1isolem1 10583 xrnegiso 11031 |
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