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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 6599 | . . 3 ⊢ (𝐵 = 𝐶 → (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻:𝐴–1-1-onto→𝐶)) | |
2 | 1 | anbi1d 629 | . 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 6357 | . 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 6357 | . 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 315 | 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 207 ∧ wa 396 = wceq 1528 ∀wral 3135 class class class wbr 5057 –1-1-onto→wf1o 6347 ‘cfv 6348 Isom wiso 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-in 3940 df-ss 3949 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-isom 6357 |
This theorem is referenced by: isores3 7077 ordiso 8968 ordtypelem9 8978 ordtypelem10 8979 oiid 8993 iunfictbso 9528 ltweuz 13317 fz1isolem 13807 dvgt0lem2 24527 erdszelem1 32335 erdsze 32346 erdsze2lem1 32347 erdsze2lem2 32348 alephiso3 39796 fourierdlem50 42318 fourierdlem89 42357 fourierdlem90 42358 fourierdlem91 42359 fourierdlem96 42364 fourierdlem97 42365 fourierdlem98 42366 fourierdlem99 42367 fourierdlem100 42368 fourierdlem108 42376 fourierdlem110 42378 fourierdlem112 42380 fourierdlem113 42381 |
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