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Mirrors > Home > MPE Home > Th. List > isoeq2 | Structured version Visualization version 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 4655 | . . . . 5 ⊢ (𝑅 = 𝑇 → (𝑥𝑅𝑦 ↔ 𝑥𝑇𝑦)) | |
2 | 1 | bibi1d 333 | . . . 4 ⊢ (𝑅 = 𝑇 → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (𝑥𝑇𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) |
3 | 2 | 2ralbidv 2989 | . . 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 740 | . 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 5897 | . 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 5897 | . 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 303 | 1 ⊢ (𝑅 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑇, 𝑆 (𝐴, 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∀wral 2912 class class class wbr 4653 –1-1-onto→wf1o 5887 ‘cfv 5888 Isom wiso 5889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-cleq 2615 df-clel 2618 df-ral 2917 df-br 4654 df-isom 5897 |
This theorem is referenced by: leiso 13243 gtiso 29478 |
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