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Mirrors > Home > MPE Home > Th. List > fsetcdmex | Structured version Visualization version GIF version |
Description: The class of all functions from a nonempty set 𝐴 into a class 𝐵 is a set iff 𝐵 is a set . (Contributed by AV, 15-Sep-2024.) |
Ref | Expression |
---|---|
fsetcdmex | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (𝐵 ∈ V ↔ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsetex 8873 | . 2 ⊢ (𝐵 ∈ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V) | |
2 | fsetprcnex 8879 | . . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V) | |
3 | 2 | ex 411 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (𝐵 ∉ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)) |
4 | df-nel 3037 | . . . 4 ⊢ (𝐵 ∉ V ↔ ¬ 𝐵 ∈ V) | |
5 | df-nel 3037 | . . . 4 ⊢ ({𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V ↔ ¬ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V) | |
6 | 3, 4, 5 | 3imtr3g 294 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (¬ 𝐵 ∈ V → ¬ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)) |
7 | 6 | con4d 115 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → ({𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V → 𝐵 ∈ V)) |
8 | 1, 7 | impbid2 225 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (𝐵 ∈ V ↔ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 {cab 2702 ≠ wne 2930 ∉ wnel 3036 Vcvv 3463 ∅c0 4318 ⟶wf 6539 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7991 df-2nd 7992 df-map 8845 |
This theorem is referenced by: (None) |
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