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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 6606 | . . 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 6364 | . 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 6364 | . 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 316 | 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 208 ∧ wa 398 = wceq 1537 ∀wral 3138 class class class wbr 5066 –1-1-onto→wf1o 6354 ‘cfv 6355 Isom wiso 6356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-in 3943 df-ss 3952 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-isom 6364 |
This theorem is referenced by: isores3 7088 ordiso 8980 ordtypelem9 8990 ordtypelem10 8991 oiid 9005 iunfictbso 9540 ltweuz 13330 fz1isolem 13820 dvgt0lem2 24600 erdszelem1 32438 erdsze 32449 erdsze2lem1 32450 erdsze2lem2 32451 alephiso3 39938 fourierdlem50 42461 fourierdlem89 42500 fourierdlem90 42501 fourierdlem91 42502 fourierdlem96 42507 fourierdlem97 42508 fourierdlem98 42509 fourierdlem99 42510 fourierdlem100 42511 fourierdlem108 42519 fourierdlem110 42521 fourierdlem112 42523 fourierdlem113 42524 |
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