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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 47662 | . 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐵 ↔ (𝐵 = ∅ ∨ ∃𝑦 𝐵 = {𝑦})) | |
2 | mof02 47667 | . . 3 ⊢ (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴⟶𝐵) | |
3 | mofsn2 47673 | . . . 4 ⊢ (𝐵 = {𝑦} → ∃*𝑓 𝑓:𝐴⟶𝐵) | |
4 | 3 | exlimiv 1932 | . . 3 ⊢ (∃𝑦 𝐵 = {𝑦} → ∃*𝑓 𝑓:𝐴⟶𝐵) |
5 | 2, 4 | jaoi 854 | . 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 844 = wceq 1540 ∃wex 1780 ∈ wcel 2105 ∃*wmo 2531 ∅c0 4322 {csn 4628 ⟶wf 6539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 |
This theorem is referenced by: setcthin 47837 |
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