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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 5399 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐹:𝐵–1-1→𝐶)) | |
2 | foeq2 5417 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–onto→𝐶 ↔ 𝐹:𝐵–onto→𝐶)) | |
3 | 1, 2 | anbi12d 470 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐹:𝐴–1-1→𝐶 ∧ 𝐹:𝐴–onto→𝐶) ↔ (𝐹:𝐵–1-1→𝐶 ∧ 𝐹:𝐵–onto→𝐶))) |
4 | df-f1o 5205 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐶 ↔ (𝐹:𝐴–1-1→𝐶 ∧ 𝐹:𝐴–onto→𝐶)) | |
5 | df-f1o 5205 | . 2 ⊢ (𝐹:𝐵–1-1-onto→𝐶 ↔ (𝐹:𝐵–1-1→𝐶 ∧ 𝐹:𝐵–onto→𝐶)) | |
6 | 3, 4, 5 | 3bitr4g 222 | 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 103 ↔ wb 104 = wceq 1348 –1-1→wf1 5195 –onto→wfo 5196 –1-1-onto→wf1o 5197 |
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 1440 ax-gen 1442 ax-4 1503 ax-17 1519 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-cleq 2163 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 |
This theorem is referenced by: f1oeq23 5434 f1oeq123d 5437 f1oeq2d 5438 f1osng 5483 isoeq4 5783 bren 6725 f1dmvrnfibi 6921 summodclem3 11343 summodclem2a 11344 summodc 11346 fsum3 11350 fsumf1o 11353 sumsnf 11372 fprodf1o 11551 prodsnf 11555 |
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