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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 5220 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵)) | |
2 | 1 | anbi1d 461 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐶) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = 𝐶))) |
3 | df-fo 5137 | . 2 ⊢ (𝐹:𝐴–onto→𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐶)) | |
4 | df-fo 5137 | . 2 ⊢ (𝐹:𝐵–onto→𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = 𝐶)) | |
5 | 2, 3, 4 | 3bitr4g 222 | 1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–onto→𝐶 ↔ 𝐹:𝐵–onto→𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ran crn 4548 Fn wfn 5126 –onto→wfo 5129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-gen 1426 ax-4 1488 ax-17 1507 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-cleq 2133 df-fn 5134 df-fo 5137 |
This theorem is referenced by: f1oeq2 5365 foeq123d 5369 tposfo 6176 ctssdclemr 7005 enumct 7008 exmidfodomrlemr 7075 exmidfodomrlemrALT 7076 ctinf 11979 ctiunct 11989 subctctexmid 13369 |
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