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| Mirrors > Home > MPE Home > Th. List > Mathboxes > f1mo | Structured version Visualization version GIF version | ||
| Description: A function that maps a set with at most one element to a class is injective. (Contributed by Zhi Wang, 1-Oct-2024.) |
| Ref | Expression |
|---|---|
| f1mo | ⊢ ((∃*𝑥 𝑥 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mo0sn 48735 | . 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦})) | |
| 2 | f102g 48761 | . . 3 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵) | |
| 3 | vex 3484 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
| 4 | f1sn2g 48760 | . . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝐹:{𝑦}⟶𝐵) → 𝐹:{𝑦}–1-1→𝐵) | |
| 5 | 3, 4 | mpan 690 | . . . . . 6 ⊢ (𝐹:{𝑦}⟶𝐵 → 𝐹:{𝑦}–1-1→𝐵) |
| 6 | feq2 6717 | . . . . . . 7 ⊢ (𝐴 = {𝑦} → (𝐹:𝐴⟶𝐵 ↔ 𝐹:{𝑦}⟶𝐵)) | |
| 7 | f1eq2 6800 | . . . . . . 7 ⊢ (𝐴 = {𝑦} → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:{𝑦}–1-1→𝐵)) | |
| 8 | 6, 7 | imbi12d 344 | . . . . . 6 ⊢ (𝐴 = {𝑦} → ((𝐹:𝐴⟶𝐵 → 𝐹:𝐴–1-1→𝐵) ↔ (𝐹:{𝑦}⟶𝐵 → 𝐹:{𝑦}–1-1→𝐵))) |
| 9 | 5, 8 | mpbiri 258 | . . . . 5 ⊢ (𝐴 = {𝑦} → (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–1-1→𝐵)) |
| 10 | 9 | exlimiv 1930 | . . . 4 ⊢ (∃𝑦 𝐴 = {𝑦} → (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–1-1→𝐵)) |
| 11 | 10 | imp 406 | . . 3 ⊢ ((∃𝑦 𝐴 = {𝑦} ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵) |
| 12 | 2, 11 | jaoian 959 | . 2 ⊢ (((𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}) ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵) |
| 13 | 1, 12 | sylanb 581 | 1 ⊢ ((∃*𝑥 𝑥 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∃*wmo 2538 Vcvv 3480 ∅c0 4333 {csn 4626 ⟶wf 6557 –1-1→wf1 6558 |
| 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-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 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-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 |
| This theorem is referenced by: thincfth 49101 |
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