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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 5261 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1→𝐴 ↔ 𝐹:𝐶–1-1→𝐵)) | |
2 | foeq3 5279 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–onto→𝐴 ↔ 𝐹:𝐶–onto→𝐵)) | |
3 | 1, 2 | anbi12d 460 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐹:𝐶–1-1→𝐴 ∧ 𝐹:𝐶–onto→𝐴) ↔ (𝐹:𝐶–1-1→𝐵 ∧ 𝐹:𝐶–onto→𝐵))) |
4 | df-f1o 5066 | . 2 ⊢ (𝐹:𝐶–1-1-onto→𝐴 ↔ (𝐹:𝐶–1-1→𝐴 ∧ 𝐹:𝐶–onto→𝐴)) | |
5 | df-f1o 5066 | . 2 ⊢ (𝐹:𝐶–1-1-onto→𝐵 ↔ (𝐹:𝐶–1-1→𝐵 ∧ 𝐹:𝐶–onto→𝐵)) | |
6 | 3, 4, 5 | 3bitr4g 222 | 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 103 ↔ wb 104 = wceq 1299 –1-1→wf1 5056 –onto→wfo 5057 –1-1-onto→wf1o 5058 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-11 1452 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-in 3027 df-ss 3034 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 |
This theorem is referenced by: f1oeq23 5295 f1oeq123d 5298 f1ores 5316 resdif 5323 f1osng 5342 f1oresrab 5517 isoeq5 5638 isoini2 5652 mapsnf1o 6561 bren 6571 xpcomf1o 6648 frechashgf1o 10042 sumeq1 10963 fisumss 11000 fsumcnv 11045 ennnfonelemhf1o 11718 ennnfonelemex 11719 |
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