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Mirrors > Home > MPE Home > Th. List > Mathboxes > mofmo | Structured version Visualization version GIF version |
Description: There is at most one function into a class containing at most one element. (Contributed by Zhi Wang, 19-Sep-2024.) |
Ref | Expression |
---|---|
mofmo | ⊢ (∃*𝑥 𝑥 ∈ 𝐵 → ∃*𝑓 𝑓:𝐴⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mo0sn 47453 | . 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐵 ↔ (𝐵 = ∅ ∨ ∃𝑦 𝐵 = {𝑦})) | |
2 | mof02 47458 | . . 3 ⊢ (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴⟶𝐵) | |
3 | mofsn2 47464 | . . . 4 ⊢ (𝐵 = {𝑦} → ∃*𝑓 𝑓:𝐴⟶𝐵) | |
4 | 3 | exlimiv 1933 | . . 3 ⊢ (∃𝑦 𝐵 = {𝑦} → ∃*𝑓 𝑓:𝐴⟶𝐵) |
5 | 2, 4 | jaoi 855 | . 2 ⊢ ((𝐵 = ∅ ∨ ∃𝑦 𝐵 = {𝑦}) → ∃*𝑓 𝑓:𝐴⟶𝐵) |
6 | 1, 5 | sylbi 216 | 1 ⊢ (∃*𝑥 𝑥 ∈ 𝐵 → ∃*𝑓 𝑓:𝐴⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 845 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∃*wmo 2532 ∅c0 4321 {csn 4627 ⟶wf 6536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 |
This theorem is referenced by: setcthin 47628 |
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