![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > dff1o6 | GIF version |
Description: A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.) |
Ref | Expression |
---|---|
dff1o6 | ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f1o 4976 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵)) | |
2 | dff13 5487 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))) | |
3 | df-fo 4975 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) | |
4 | 2, 3 | anbi12i 448 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵) ↔ ((𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))) |
5 | df-3an 922 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))) | |
6 | eqimss 3062 | . . . . . . 7 ⊢ (ran 𝐹 = 𝐵 → ran 𝐹 ⊆ 𝐵) | |
7 | 6 | anim2i 334 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) |
8 | df-f 4973 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
9 | 7, 8 | sylibr 132 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴⟶𝐵) |
10 | 9 | pm4.71ri 384 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))) |
11 | 10 | anbi1i 446 | . . 3 ⊢ (((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) ↔ ((𝐹:𝐴⟶𝐵 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))) |
12 | an32 527 | . . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) ↔ ((𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))) | |
13 | 5, 11, 12 | 3bitrri 205 | . 2 ⊢ (((𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))) |
14 | 1, 4, 13 | 3bitri 204 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∧ w3a 920 = wceq 1285 ∀wral 2353 ⊆ wss 2984 ran crn 4402 Fn wfn 4964 ⟶wf 4965 –1-1→wf1 4966 –onto→wfo 4967 –1-1-onto→wf1o 4968 ‘cfv 4969 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2614 df-sbc 2827 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-id 4084 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-f1 4974 df-fo 4975 df-f1o 4976 df-fv 4977 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |