![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > foeq2 | GIF version |
Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) |
Ref | Expression |
---|---|
foeq2 | ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–onto→𝐶 ↔ 𝐹:𝐵–onto→𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fneq2 5306 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵)) | |
2 | 1 | anbi1d 465 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐶) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = 𝐶))) |
3 | df-fo 5223 | . 2 ⊢ (𝐹:𝐴–onto→𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐶)) | |
4 | df-fo 5223 | . 2 ⊢ (𝐹:𝐵–onto→𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = 𝐶)) | |
5 | 2, 3, 4 | 3bitr4g 223 | 1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–onto→𝐶 ↔ 𝐹:𝐵–onto→𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ran crn 4628 Fn wfn 5212 –onto→wfo 5215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-4 1510 ax-17 1526 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-cleq 2170 df-fn 5220 df-fo 5223 |
This theorem is referenced by: f1oeq2 5451 foeq123d 5455 tposfo 6272 ctssdclemr 7111 enumct 7114 exmidfodomrlemr 7201 exmidfodomrlemrALT 7202 ctinf 12431 ctiunct 12441 ssomct 12446 subctctexmid 14753 |
Copyright terms: Public domain | W3C validator |