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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 7247 | . 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 2933 | . . . . . . 7 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦) | |
4 | 3 | bicomi 223 | . . . . . 6 ⊢ (¬ 𝑥 = 𝑦 ↔ 𝑥 ≠ 𝑦) |
5 | df-ne 2933 | . . . . . . 7 ⊢ ((𝐹‘𝑥) ≠ (𝐹‘𝑦) ↔ ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) | |
6 | 5 | bicomi 223 | . . . . . 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 3120 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐹‘𝑥) ≠ (𝐹‘𝑦))) |
10 | 9 | anbi2i 622 | . 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 205 ∧ wa 395 = wceq 1533 ≠ wne 2932 ∀wral 3053 ⟶wf 6530 –1-1→wf1 6531 ‘cfv 6534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fv 6542 |
This theorem is referenced by: dff14b 7263 pthdlem1 29518 fldhmf1 41462 nnfoctbdjlem 45717 |
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