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Mirrors > Home > ILE Home > Th. List > isoeq2 | GIF version |
Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Ref | Expression |
---|---|
isoeq2 | ⊢ (𝑅 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑇, 𝑆 (𝐴, 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq 3766 | . . . . 5 ⊢ (𝑅 = 𝑇 → (𝑥𝑅𝑦 ↔ 𝑥𝑇𝑦)) | |
2 | 1 | bibi1d 222 | . . . 4 ⊢ (𝑅 = 𝑇 → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (𝑥𝑇𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) |
3 | 2 | 2ralbidv 2348 | . . 3 ⊢ (𝑅 = 𝑇 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑇𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) |
4 | 3 | anbi2d 437 | . 2 ⊢ (𝑅 = 𝑇 → ((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑇𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))) |
5 | df-isom 4911 | . 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) | |
6 | df-isom 4911 | . 2 ⊢ (𝐻 Isom 𝑇, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑇𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) | |
7 | 4, 5, 6 | 3bitr4g 212 | 1 ⊢ (𝑅 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑇, 𝑆 (𝐴, 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∀wral 2306 class class class wbr 3764 –1-1-onto→wf1o 4901 ‘cfv 4902 Isom wiso 4903 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-17 1419 ax-ial 1427 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-cleq 2033 df-clel 2036 df-ral 2311 df-br 3765 df-isom 4911 |
This theorem is referenced by: (None) |
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