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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 6775 | . . 3 ⊢ (𝐵 = 𝐶 → (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻:𝐴–1-1-onto→𝐶)) | |
2 | 1 | anbi1d 631 | . 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 6506 | . 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 6506 | . 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 314 | 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 205 ∧ wa 397 = wceq 1542 ∀wral 3061 class class class wbr 5106 –1-1-onto→wf1o 6496 ‘cfv 6497 Isom wiso 6498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3446 df-in 3918 df-ss 3928 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-isom 6506 |
This theorem is referenced by: isores3 7281 ordiso 9457 ordtypelem9 9467 ordtypelem10 9468 oiid 9482 iunfictbso 10055 ltweuz 13872 fz1isolem 14366 dvgt0lem2 25383 erdszelem1 33842 erdsze 33853 erdsze2lem1 33854 erdsze2lem2 33855 isoeq145d 41779 alephiso3 41919 fourierdlem50 44483 fourierdlem89 44522 fourierdlem90 44523 fourierdlem91 44524 fourierdlem96 44529 fourierdlem97 44530 fourierdlem98 44531 fourierdlem99 44532 fourierdlem100 44533 fourierdlem108 44541 fourierdlem110 44543 fourierdlem112 44545 fourierdlem113 44546 |
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