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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 2758 | . . . . . . 7 ⊢ {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {〈𝑆, 𝑏〉}} = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {〈𝑆, 𝑏〉}} | |
2 | eqid 2758 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ {〈𝑆, 𝑥〉}) = (𝑥 ∈ 𝐵 ↦ {〈𝑆, 𝑥〉}) | |
3 | 1, 2 | fsetsnf1o 44056 | . . . . . 6 ⊢ (𝑆 ∈ 𝑉 → (𝑥 ∈ 𝐵 ↦ {〈𝑆, 𝑥〉}):𝐵–1-1-onto→{𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {〈𝑆, 𝑏〉}}) |
4 | f1ovv 7669 | . . . . . 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 3056 | . . . 4 ⊢ (𝐵 ∉ V ↔ ¬ 𝐵 ∈ V) | |
8 | df-nel 3056 | . . . 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 44052 | . . . 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 2759 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → V = V) | |
14 | 12, 13 | neleq12d 3059 | . 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 1538 ∈ wcel 2111 {cab 2735 ∉ wnel 3055 ∃wrex 3071 Vcvv 3409 {csn 4525 〈cop 4531 ↦ cmpt 5116 ⟶wf 6336 –1-1-onto→wf1o 6339 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pr 5302 ax-un 7465 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-id 5434 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 |
This theorem is referenced by: fsetprcnexALT 44064 |
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