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Mirrors > Home > MPE Home > Th. List > dff14b | 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 |
---|---|
dff14b | ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹‘𝑥) ≠ (𝐹‘𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dff14a 6755 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐹‘𝑥) ≠ (𝐹‘𝑦)))) | |
2 | necom 3024 | . . . . . . 7 ⊢ (𝑥 ≠ 𝑦 ↔ 𝑦 ≠ 𝑥) | |
3 | 2 | imbi1i 341 | . . . . . 6 ⊢ ((𝑥 ≠ 𝑦 → (𝐹‘𝑥) ≠ (𝐹‘𝑦)) ↔ (𝑦 ≠ 𝑥 → (𝐹‘𝑥) ≠ (𝐹‘𝑦))) |
4 | 3 | ralbii 3161 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐹‘𝑥) ≠ (𝐹‘𝑦)) ↔ ∀𝑦 ∈ 𝐴 (𝑦 ≠ 𝑥 → (𝐹‘𝑥) ≠ (𝐹‘𝑦))) |
5 | raldifsnb 4515 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝑦 ≠ 𝑥 → (𝐹‘𝑥) ≠ (𝐹‘𝑦)) ↔ ∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹‘𝑥) ≠ (𝐹‘𝑦)) | |
6 | 4, 5 | bitri 267 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐹‘𝑥) ≠ (𝐹‘𝑦)) ↔ ∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹‘𝑥) ≠ (𝐹‘𝑦)) |
7 | 6 | ralbii 3161 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐹‘𝑥) ≠ (𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹‘𝑥) ≠ (𝐹‘𝑦)) |
8 | 7 | anbi2i 617 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐹‘𝑥) ≠ (𝐹‘𝑦))) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹‘𝑥) ≠ (𝐹‘𝑦))) |
9 | 1, 8 | bitri 267 | 1 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹‘𝑥) ≠ (𝐹‘𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ≠ wne 2971 ∀wral 3089 ∖ cdif 3766 {csn 4368 ⟶wf 6097 –1-1→wf1 6098 ‘cfv 6101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fv 6109 |
This theorem is referenced by: f12dfv 6757 f13dfv 6758 |
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