![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > f1oeq1 | GIF version |
Description: Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.) |
Ref | Expression |
---|---|
f1oeq1 | ⊢ (𝐹 = 𝐺 → (𝐹:A–1-1-onto→B ↔ 𝐺:A–1-1-onto→B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1eq1 5030 | . . 3 ⊢ (𝐹 = 𝐺 → (𝐹:A–1-1→B ↔ 𝐺:A–1-1→B)) | |
2 | foeq1 5045 | . . 3 ⊢ (𝐹 = 𝐺 → (𝐹:A–onto→B ↔ 𝐺:A–onto→B)) | |
3 | 1, 2 | anbi12d 442 | . 2 ⊢ (𝐹 = 𝐺 → ((𝐹:A–1-1→B ∧ 𝐹:A–onto→B) ↔ (𝐺:A–1-1→B ∧ 𝐺:A–onto→B))) |
4 | df-f1o 4852 | . 2 ⊢ (𝐹:A–1-1-onto→B ↔ (𝐹:A–1-1→B ∧ 𝐹:A–onto→B)) | |
5 | df-f1o 4852 | . 2 ⊢ (𝐺:A–1-1-onto→B ↔ (𝐺:A–1-1→B ∧ 𝐺:A–onto→B)) | |
6 | 3, 4, 5 | 3bitr4g 212 | 1 ⊢ (𝐹 = 𝐺 → (𝐹:A–1-1-onto→B ↔ 𝐺:A–1-1-onto→B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 –1-1→wf1 4842 –onto→wfo 4843 –1-1-onto→wf1o 4844 |
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-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 |
This theorem is referenced by: f1oeq123d 5066 f1ocnvb 5083 f1orescnv 5085 f1ovi 5108 f1osng 5110 f1oresrab 5272 fsn 5278 isoeq1 5384 f1oen3g 6170 ensn1 6212 xpcomf1o 6235 |
Copyright terms: Public domain | W3C validator |