![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > isoeq1 | Structured version Visualization version GIF version |
Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
Ref | Expression |
---|---|
isoeq1 | ⊢ (𝐻 = 𝐺 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1oeq1 6579 | . . 3 ⊢ (𝐻 = 𝐺 → (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐺:𝐴–1-1-onto→𝐵)) | |
2 | fveq1 6644 | . . . . . 6 ⊢ (𝐻 = 𝐺 → (𝐻‘𝑥) = (𝐺‘𝑥)) | |
3 | fveq1 6644 | . . . . . 6 ⊢ (𝐻 = 𝐺 → (𝐻‘𝑦) = (𝐺‘𝑦)) | |
4 | 2, 3 | breq12d 5043 | . . . . 5 ⊢ (𝐻 = 𝐺 → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐺‘𝑥)𝑆(𝐺‘𝑦))) |
5 | 4 | bibi2d 346 | . . . 4 ⊢ (𝐻 = 𝐺 → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (𝑥𝑅𝑦 ↔ (𝐺‘𝑥)𝑆(𝐺‘𝑦)))) |
6 | 5 | 2ralbidv 3164 | . . 3 ⊢ (𝐻 = 𝐺 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐺‘𝑥)𝑆(𝐺‘𝑦)))) |
7 | 1, 6 | anbi12d 633 | . 2 ⊢ (𝐻 = 𝐺 → ((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ↔ (𝐺:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐺‘𝑥)𝑆(𝐺‘𝑦))))) |
8 | df-isom 6333 | . 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) | |
9 | df-isom 6333 | . 2 ⊢ (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐺:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐺‘𝑥)𝑆(𝐺‘𝑦)))) | |
10 | 7, 8, 9 | 3bitr4g 317 | 1 ⊢ (𝐻 = 𝐺 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∀wral 3106 class class class wbr 5030 –1-1-onto→wf1o 6323 ‘cfv 6324 Isom wiso 6325 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 |
This theorem is referenced by: isores1 7066 wemoiso 7656 wemoiso2 7657 ordiso 8964 oieu 8987 finnisoeu 9524 iunfictbso 9525 infrenegsup 11611 ltweuz 13324 fz1isolem 13815 isercolllem2 15014 isercoll 15016 dvgt0lem2 24606 efcvx 25044 relogiso 25189 logccv 25254 erdszelem1 32551 erdsze 32562 erdsze2lem2 32564 fzisoeu 41932 fourierdlem36 42785 fourierdlem96 42844 fourierdlem97 42845 fourierdlem98 42846 fourierdlem99 42847 fourierdlem105 42853 fourierdlem106 42854 fourierdlem108 42856 fourierdlem110 42858 fourierdlem112 42860 fourierdlem113 42861 fourierdlem115 42863 rrx2plordisom 45137 |
Copyright terms: Public domain | W3C validator |