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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 8796 for 𝐴 ∉ V, fset0 8795 for 𝐴 = ∅, and fsetex 8797 for 𝐵 ∈ V, see also fsetexb 8805. (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 4307 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑎 𝑎 ∈ 𝐴) | |
2 | feq1 6650 | . . . . . . . . . 10 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴⟶𝐵 ↔ 𝑚:𝐴⟶𝐵)) | |
3 | 2 | cbvabv 2806 | . . . . . . . . 9 ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} = {𝑚 ∣ 𝑚:𝐴⟶𝐵} |
4 | fveq1 6842 | . . . . . . . . . 10 ⊢ (𝑔 = 𝑛 → (𝑔‘𝑎) = (𝑛‘𝑎)) | |
5 | 4 | cbvmptv 5219 | . . . . . . . . 9 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↦ (𝑔‘𝑎)) = (𝑛 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↦ (𝑛‘𝑎)) |
6 | 3, 5 | fsetfocdm 8802 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴) → (𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↦ (𝑔‘𝑎)):{𝑓 ∣ 𝑓:𝐴⟶𝐵}–onto→𝐵) |
7 | focdmex 7889 | . . . . . . . 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 3058 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴) → (𝐵 ∉ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)) |
10 | 9 | expcom 415 | . . . . 5 ⊢ (𝑎 ∈ 𝐴 → (𝐴 ∈ 𝑉 → (𝐵 ∉ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V))) |
11 | 10 | exlimiv 1934 | . . . 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 409 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (𝐵 ∉ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)) |
14 | 13 | imp 408 | 1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∃wex 1782 ∈ wcel 2107 {cab 2710 ≠ wne 2940 ∉ wnel 3046 Vcvv 3444 ∅c0 4283 ↦ cmpt 5189 ⟶wf 6493 –onto→wfo 6495 ‘cfv 6497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 |
This theorem is referenced by: fsetcdmex 8804 fsetexb 8805 |
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