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Mirrors > Home > MPE Home > Th. List > isoeq5 | Structured version Visualization version GIF version |
Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Ref | Expression |
---|---|
isoeq5 | ⊢ (𝐵 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1oeq3 6129 | . . 3 ⊢ (𝐵 = 𝐶 → (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻:𝐴–1-1-onto→𝐶)) | |
2 | 1 | anbi1d 741 | . 2 ⊢ (𝐵 = 𝐶 → ((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ↔ (𝐻:𝐴–1-1-onto→𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))) |
3 | 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)))) | |
4 | df-isom 5897 | . 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐶) ↔ (𝐻:𝐴–1-1-onto→𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) | |
5 | 2, 3, 4 | 3bitr4g 303 | 1 ⊢ (𝐵 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐶))) |
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-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-in 3581 df-ss 3588 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-isom 5897 |
This theorem is referenced by: isores3 6585 ordiso 8421 ordtypelem9 8431 ordtypelem10 8432 oiid 8446 iunfictbso 8937 ltweuz 12760 fz1isolem 13245 dvgt0lem2 23766 erdszelem1 31173 erdsze 31184 erdsze2lem1 31185 erdsze2lem2 31186 fourierdlem50 40373 fourierdlem89 40412 fourierdlem90 40413 fourierdlem91 40414 fourierdlem96 40419 fourierdlem97 40420 fourierdlem98 40421 fourierdlem99 40422 fourierdlem100 40423 fourierdlem108 40431 fourierdlem110 40433 fourierdlem112 40435 fourierdlem113 40436 |
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