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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 5324 | . . 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 5241 | . 2 ⊢ (𝐹:𝐴–onto→𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐶)) | |
4 | df-fo 5241 | . 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 1364 ran crn 4645 Fn wfn 5230 –onto→wfo 5233 |
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 1458 ax-gen 1460 ax-4 1521 ax-17 1537 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-cleq 2182 df-fn 5238 df-fo 5241 |
This theorem is referenced by: f1oeq2 5469 foeq123d 5473 tposfo 6295 ctssdclemr 7140 enumct 7143 exmidfodomrlemr 7230 exmidfodomrlemrALT 7231 ctinf 12480 ctiunct 12490 ssomct 12495 subctctexmid 15204 |
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