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| Mirrors > Home > MPE Home > Th. List > dff14a | Structured version Visualization version GIF version | ||
| Description: A one-to-one function in terms of different function values for different arguments. (Contributed by Alexander van der Vekens, 26-Jan-2018.) | 
| Ref | Expression | 
|---|---|
| dff14a | ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐹‘𝑥) ≠ (𝐹‘𝑦)))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dff13 7275 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))) | |
| 2 | con34b 316 | . . . . 5 ⊢ (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ (¬ 𝑥 = 𝑦 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) | |
| 3 | df-ne 2941 | . . . . . . 7 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦) | |
| 4 | 3 | bicomi 224 | . . . . . 6 ⊢ (¬ 𝑥 = 𝑦 ↔ 𝑥 ≠ 𝑦) | 
| 5 | df-ne 2941 | . . . . . . 7 ⊢ ((𝐹‘𝑥) ≠ (𝐹‘𝑦) ↔ ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) | |
| 6 | 5 | bicomi 224 | . . . . . 6 ⊢ (¬ (𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑥) ≠ (𝐹‘𝑦)) | 
| 7 | 4, 6 | imbi12i 350 | . . . . 5 ⊢ ((¬ 𝑥 = 𝑦 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ (𝑥 ≠ 𝑦 → (𝐹‘𝑥) ≠ (𝐹‘𝑦))) | 
| 8 | 2, 7 | bitri 275 | . . . 4 ⊢ (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ (𝑥 ≠ 𝑦 → (𝐹‘𝑥) ≠ (𝐹‘𝑦))) | 
| 9 | 8 | 2ralbii 3128 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐹‘𝑥) ≠ (𝐹‘𝑦))) | 
| 10 | 9 | anbi2i 623 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐹‘𝑥) ≠ (𝐹‘𝑦)))) | 
| 11 | 1, 10 | bitri 275 | 1 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐹‘𝑥) ≠ (𝐹‘𝑦)))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ≠ wne 2940 ∀wral 3061 ⟶wf 6557 –1-1→wf1 6558 ‘cfv 6561 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fv 6569 | 
| This theorem is referenced by: dff14b 7291 f1ounsn 7292 resf1extb 7956 pthdlem1 29786 fldhmf1 42091 nnfoctbdjlem 46470 isubgr3stgrlem4 47936 | 
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