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Mirrors > Home > MPE Home > Th. List > Mathboxes > fsetsnprcnex | Structured version Visualization version GIF version |
Description: The class of all functions from a (proper) singleton into a proper class 𝐵 is not a set. (Contributed by AV, 13-Sep-2024.) |
Ref | Expression |
---|---|
fsetsnprcnex | ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} ∉ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . . . . 7 ⊢ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {〈𝑆, 𝑏〉}} = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {〈𝑆, 𝑏〉}} | |
2 | eqid 2737 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ {〈𝑆, 𝑥〉}) = (𝑥 ∈ 𝐵 ↦ {〈𝑆, 𝑥〉}) | |
3 | 1, 2 | fsetsnf1o 44228 | . . . . . 6 ⊢ (𝑆 ∈ 𝑉 → (𝑥 ∈ 𝐵 ↦ {〈𝑆, 𝑥〉}):𝐵–1-1-onto→{𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {〈𝑆, 𝑏〉}}) |
4 | f1ovv 7736 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ↦ {〈𝑆, 𝑥〉}):𝐵–1-1-onto→{𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {〈𝑆, 𝑏〉}} → (𝐵 ∈ V ↔ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {〈𝑆, 𝑏〉}} ∈ V)) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝑆 ∈ 𝑉 → (𝐵 ∈ V ↔ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {〈𝑆, 𝑏〉}} ∈ V)) |
6 | 5 | notbid 321 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → (¬ 𝐵 ∈ V ↔ ¬ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {〈𝑆, 𝑏〉}} ∈ V)) |
7 | df-nel 3047 | . . . 4 ⊢ (𝐵 ∉ V ↔ ¬ 𝐵 ∈ V) | |
8 | df-nel 3047 | . . . 4 ⊢ ({𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {〈𝑆, 𝑏〉}} ∉ V ↔ ¬ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {〈𝑆, 𝑏〉}} ∈ V) | |
9 | 6, 7, 8 | 3bitr4g 317 | . . 3 ⊢ (𝑆 ∈ 𝑉 → (𝐵 ∉ V ↔ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {〈𝑆, 𝑏〉}} ∉ V)) |
10 | 9 | biimpa 480 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {〈𝑆, 𝑏〉}} ∉ V) |
11 | fsetabsnop 44224 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {〈𝑆, 𝑏〉}}) | |
12 | 11 | adantr 484 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {〈𝑆, 𝑏〉}}) |
13 | eqidd 2738 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → V = V) | |
14 | 12, 13 | neleq12d 3050 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → ({𝑓 ∣ 𝑓:{𝑆}⟶𝐵} ∉ V ↔ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {〈𝑆, 𝑏〉}} ∉ V)) |
15 | 10, 14 | mpbird 260 | 1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} ∉ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 {cab 2714 ∉ wnel 3046 ∃wrex 3062 Vcvv 3413 {csn 4546 〈cop 4552 ↦ cmpt 5140 ⟶wf 6381 –1-1-onto→wf1o 6384 |
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 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5184 ax-sep 5197 ax-nul 5204 ax-pr 5327 ax-un 7528 |
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 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3415 df-sbc 3700 df-csb 3817 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-nul 4243 df-if 4445 df-sn 4547 df-pr 4549 df-op 4553 df-uni 4825 df-iun 4911 df-br 5059 df-opab 5121 df-mpt 5141 df-id 5460 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-rn 5567 df-res 5568 df-ima 5569 df-iota 6343 df-fun 6387 df-fn 6388 df-f 6389 df-f1 6390 df-fo 6391 df-f1o 6392 df-fv 6393 |
This theorem is referenced by: fsetprcnexALT 44236 |
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