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Mathbox for Zhi Wang |
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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 48664 | . 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦})) | |
2 | f102g 48682 | . . 3 ⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵) | |
3 | vex 3482 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
4 | f1sn2g 48681 | . . . . . . 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 6718 | . . . . . . 7 ⊢ (𝐴 = {𝑦} → (𝐹:𝐴⟶𝐵 ↔ 𝐹:{𝑦}⟶𝐵)) | |
7 | f1eq2 6801 | . . . . . . 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 1928 | . . . 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 958 | . 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 847 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ∃*wmo 2536 Vcvv 3478 ∅c0 4339 {csn 4631 ⟶wf 6559 –1-1→wf1 6560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 |
This theorem is referenced by: thincfth 48848 |
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