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Mirrors > Home > MPE Home > Th. List > Mathboxes > mofsn2 | Structured version Visualization version GIF version |
Description: There is at most one function into a singleton. An unconditional variant of mofsn 45787, i.e., the singleton could be empty if 𝑌 is a proper class. (Contributed by Zhi Wang, 19-Sep-2024.) |
Ref | Expression |
---|---|
mofsn2 | ⊢ (𝐵 = {𝑌} → ∃*𝑓 𝑓:𝐴⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mofsn 45787 | . . . 4 ⊢ (𝑌 ∈ V → ∃*𝑓 𝑓:𝐴⟶{𝑌}) | |
2 | 1 | adantl 485 | . . 3 ⊢ ((𝐵 = {𝑌} ∧ 𝑌 ∈ V) → ∃*𝑓 𝑓:𝐴⟶{𝑌}) |
3 | feq3 6506 | . . . . 5 ⊢ (𝐵 = {𝑌} → (𝑓:𝐴⟶𝐵 ↔ 𝑓:𝐴⟶{𝑌})) | |
4 | 3 | mobidv 2548 | . . . 4 ⊢ (𝐵 = {𝑌} → (∃*𝑓 𝑓:𝐴⟶𝐵 ↔ ∃*𝑓 𝑓:𝐴⟶{𝑌})) |
5 | 4 | adantr 484 | . . 3 ⊢ ((𝐵 = {𝑌} ∧ 𝑌 ∈ V) → (∃*𝑓 𝑓:𝐴⟶𝐵 ↔ ∃*𝑓 𝑓:𝐴⟶{𝑌})) |
6 | 2, 5 | mpbird 260 | . 2 ⊢ ((𝐵 = {𝑌} ∧ 𝑌 ∈ V) → ∃*𝑓 𝑓:𝐴⟶𝐵) |
7 | simpl 486 | . . . 4 ⊢ ((𝐵 = {𝑌} ∧ ¬ 𝑌 ∈ V) → 𝐵 = {𝑌}) | |
8 | snprc 4619 | . . . . . 6 ⊢ (¬ 𝑌 ∈ V ↔ {𝑌} = ∅) | |
9 | 8 | biimpi 219 | . . . . 5 ⊢ (¬ 𝑌 ∈ V → {𝑌} = ∅) |
10 | 9 | adantl 485 | . . . 4 ⊢ ((𝐵 = {𝑌} ∧ ¬ 𝑌 ∈ V) → {𝑌} = ∅) |
11 | 7, 10 | eqtrd 2771 | . . 3 ⊢ ((𝐵 = {𝑌} ∧ ¬ 𝑌 ∈ V) → 𝐵 = ∅) |
12 | mof02 45782 | . . 3 ⊢ (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴⟶𝐵) | |
13 | 11, 12 | syl 17 | . 2 ⊢ ((𝐵 = {𝑌} ∧ ¬ 𝑌 ∈ V) → ∃*𝑓 𝑓:𝐴⟶𝐵) |
14 | 6, 13 | pm2.61dan 813 | 1 ⊢ (𝐵 = {𝑌} → ∃*𝑓 𝑓:𝐴⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∃*wmo 2537 Vcvv 3398 ∅c0 4223 {csn 4527 ⟶wf 6354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 |
This theorem is referenced by: mofsssn 45789 mofmo 45790 |
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