![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fsetprcnex | Structured version Visualization version GIF version |
Description: The class of all functions from a nonempty set 𝐴 into a proper class 𝐵 is not a set. If one of the preconditions is not fufilled, then {𝑓 ∣ 𝑓:𝐴⟶𝐵} is a set, see fsetdmprc0 8880 for 𝐴 ∉ V, fset0 8879 for 𝐴 = ∅, and fsetex 8881 for 𝐵 ∈ V, see also fsetexb 8889. (Contributed by AV, 14-Sep-2024.) (Proof shortened by BJ, 15-Sep-2024.) |
Ref | Expression |
---|---|
fsetprcnex | ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4350 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑎 𝑎 ∈ 𝐴) | |
2 | feq1 6708 | . . . . . . . . . 10 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴⟶𝐵 ↔ 𝑚:𝐴⟶𝐵)) | |
3 | 2 | cbvabv 2801 | . . . . . . . . 9 ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} = {𝑚 ∣ 𝑚:𝐴⟶𝐵} |
4 | fveq1 6901 | . . . . . . . . . 10 ⊢ (𝑔 = 𝑛 → (𝑔‘𝑎) = (𝑛‘𝑎)) | |
5 | 4 | cbvmptv 5265 | . . . . . . . . 9 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↦ (𝑔‘𝑎)) = (𝑛 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↦ (𝑛‘𝑎)) |
6 | 3, 5 | fsetfocdm 8886 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴) → (𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↦ (𝑔‘𝑎)):{𝑓 ∣ 𝑓:𝐴⟶𝐵}–onto→𝐵) |
7 | focdmex 7965 | . . . . . . . 8 ⊢ ({𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V → ((𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↦ (𝑔‘𝑎)):{𝑓 ∣ 𝑓:𝐴⟶𝐵}–onto→𝐵 → 𝐵 ∈ V)) | |
8 | 6, 7 | syl5com 31 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴) → ({𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V → 𝐵 ∈ V)) |
9 | 8 | nelcon3d 3047 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴) → (𝐵 ∉ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)) |
10 | 9 | expcom 412 | . . . . 5 ⊢ (𝑎 ∈ 𝐴 → (𝐴 ∈ 𝑉 → (𝐵 ∉ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V))) |
11 | 10 | exlimiv 1925 | . . . 4 ⊢ (∃𝑎 𝑎 ∈ 𝐴 → (𝐴 ∈ 𝑉 → (𝐵 ∉ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V))) |
12 | 1, 11 | sylbi 216 | . . 3 ⊢ (𝐴 ≠ ∅ → (𝐴 ∈ 𝑉 → (𝐵 ∉ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V))) |
13 | 12 | impcom 406 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (𝐵 ∉ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)) |
14 | 13 | imp 405 | 1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∃wex 1773 ∈ wcel 2098 {cab 2705 ≠ wne 2937 ∉ wnel 3043 Vcvv 3473 ∅c0 4326 ↦ cmpt 5235 ⟶wf 6549 –onto→wfo 6551 ‘cfv 6553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 |
This theorem is referenced by: fsetcdmex 8888 fsetexb 8889 |
Copyright terms: Public domain | W3C validator |