![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > isoeq1 | 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 5060 | . . 3 ⊢ (𝐻 = 𝐺 → (𝐻:A–1-1-onto→B ↔ 𝐺:A–1-1-onto→B)) | |
2 | fveq1 5120 | . . . . . 6 ⊢ (𝐻 = 𝐺 → (𝐻‘x) = (𝐺‘x)) | |
3 | fveq1 5120 | . . . . . 6 ⊢ (𝐻 = 𝐺 → (𝐻‘y) = (𝐺‘y)) | |
4 | 2, 3 | breq12d 3768 | . . . . 5 ⊢ (𝐻 = 𝐺 → ((𝐻‘x)𝑆(𝐻‘y) ↔ (𝐺‘x)𝑆(𝐺‘y))) |
5 | 4 | bibi2d 221 | . . . 4 ⊢ (𝐻 = 𝐺 → ((x𝑅y ↔ (𝐻‘x)𝑆(𝐻‘y)) ↔ (x𝑅y ↔ (𝐺‘x)𝑆(𝐺‘y)))) |
6 | 5 | 2ralbidv 2342 | . . 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 442 | . 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 4854 | . 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 4854 | . 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 212 | 1 ⊢ (𝐻 = 𝐺 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐺 Isom 𝑅, 𝑆 (A, B))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∀wral 2300 class class class wbr 3755 –1-1-onto→wf1o 4844 ‘cfv 4845 Isom wiso 4846 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-isom 4854 |
This theorem is referenced by: isores1 5397 |
Copyright terms: Public domain | W3C validator |