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Mirrors > Home > ILE Home > Th. List > f1oeq2 | GIF version |
Description: Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.) |
Ref | Expression |
---|---|
f1oeq2 | ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1eq2 5417 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐹:𝐵–1-1→𝐶)) | |
2 | foeq2 5435 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–onto→𝐶 ↔ 𝐹:𝐵–onto→𝐶)) | |
3 | 1, 2 | anbi12d 473 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐹:𝐴–1-1→𝐶 ∧ 𝐹:𝐴–onto→𝐶) ↔ (𝐹:𝐵–1-1→𝐶 ∧ 𝐹:𝐵–onto→𝐶))) |
4 | df-f1o 5223 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐶 ↔ (𝐹:𝐴–1-1→𝐶 ∧ 𝐹:𝐴–onto→𝐶)) | |
5 | df-f1o 5223 | . 2 ⊢ (𝐹:𝐵–1-1-onto→𝐶 ↔ (𝐹:𝐵–1-1→𝐶 ∧ 𝐹:𝐵–onto→𝐶)) | |
6 | 3, 4, 5 | 3bitr4g 223 | 1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 –1-1→wf1 5213 –onto→wfo 5214 –1-1-onto→wf1o 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 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 |
This theorem is referenced by: f1oeq23 5452 f1oeq123d 5455 f1oeq2d 5457 f1osng 5502 isoeq4 5804 bren 6746 f1dmvrnfibi 6942 summodclem3 11383 summodclem2a 11384 summodc 11386 fsum3 11390 fsumf1o 11393 sumsnf 11412 fprodf1o 11591 prodsnf 11595 |
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