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Mirrors > Home > ILE Home > Th. List > f1oeq3 | GIF version |
Description: Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.) |
Ref | Expression |
---|---|
f1oeq3 | ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:𝐶–1-1-onto→𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1eq3 5089 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1→𝐴 ↔ 𝐹:𝐶–1-1→𝐵)) | |
2 | foeq3 5104 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–onto→𝐴 ↔ 𝐹:𝐶–onto→𝐵)) | |
3 | 1, 2 | anbi12d 442 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐹:𝐶–1-1→𝐴 ∧ 𝐹:𝐶–onto→𝐴) ↔ (𝐹:𝐶–1-1→𝐵 ∧ 𝐹:𝐶–onto→𝐵))) |
4 | df-f1o 4909 | . 2 ⊢ (𝐹:𝐶–1-1-onto→𝐴 ↔ (𝐹:𝐶–1-1→𝐴 ∧ 𝐹:𝐶–onto→𝐴)) | |
5 | df-f1o 4909 | . 2 ⊢ (𝐹:𝐶–1-1-onto→𝐵 ↔ (𝐹:𝐶–1-1→𝐵 ∧ 𝐹:𝐶–onto→𝐵)) | |
6 | 3, 4, 5 | 3bitr4g 212 | 1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:𝐶–1-1-onto→𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 –1-1→wf1 4899 –onto→wfo 4900 –1-1-onto→wf1o 4901 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-in 2924 df-ss 2931 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 |
This theorem is referenced by: f1oeq23 5120 f1oeq123d 5123 f1ores 5141 resdif 5148 f1osng 5167 f1oresrab 5329 isoeq5 5445 isoini2 5458 bren 6228 xpcomf1o 6299 frechashgf1o 9205 sumeq1 9874 |
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