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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 5216 | . 2 wff 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) |
7 | 1, 2, 5 | wf1o 5214 | . . 3 wff 𝐻:𝐴–1-1-onto→𝐵 |
8 | vx | . . . . . . . 8 setvar 𝑥 | |
9 | 8 | cv 1352 | . . . . . . 7 class 𝑥 |
10 | vy | . . . . . . . 8 setvar 𝑦 | |
11 | 10 | cv 1352 | . . . . . . 7 class 𝑦 |
12 | 9, 11, 3 | wbr 4002 | . . . . . 6 wff 𝑥𝑅𝑦 |
13 | 9, 5 | cfv 5215 | . . . . . . 7 class (𝐻‘𝑥) |
14 | 11, 5 | cfv 5215 | . . . . . . 7 class (𝐻‘𝑦) |
15 | 13, 14, 4 | wbr 4002 | . . . . . 6 wff (𝐻‘𝑥)𝑆(𝐻‘𝑦) |
16 | 12, 15 | wb 105 | . . . . 5 wff (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) |
17 | 16, 10, 1 | wral 2455 | . . . 4 wff ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) |
18 | 17, 8, 1 | wral 2455 | . . 3 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) |
19 | 7, 18 | wa 104 | . 2 wff (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) |
20 | 6, 19 | wb 105 | 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 5799 isoeq2 5800 isoeq3 5801 isoeq4 5802 isoeq5 5803 nfiso 5804 isof1o 5805 isorel 5806 isoid 5808 isocnv 5809 isocnv2 5810 isores2 5811 isores3 5813 isotr 5814 iso0 5815 isoini2 5817 f1oiso 5824 negiso 8908 frec2uzisod 10402 zfz1isolem1 10813 xrnegiso 11263 reefiso 14069 logltb 14166 |
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