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Mirrors > Home > MPE Home > Th. List > 0map0sn0 | Structured version Visualization version GIF version |
Description: The set of mappings of the empty set to the empty set is the singleton containing the empty set. (Contributed by AV, 31-Mar-2024.) |
Ref | Expression |
---|---|
0map0sn0 | ⊢ (∅ ↑m ∅) = {∅} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f0bi 6771 | . . 3 ⊢ (𝑓:∅⟶∅ ↔ 𝑓 = ∅) | |
2 | 1 | abbii 2802 | . 2 ⊢ {𝑓 ∣ 𝑓:∅⟶∅} = {𝑓 ∣ 𝑓 = ∅} |
3 | 0ex 5306 | . . 3 ⊢ ∅ ∈ V | |
4 | 3, 3 | mapval 8828 | . 2 ⊢ (∅ ↑m ∅) = {𝑓 ∣ 𝑓:∅⟶∅} |
5 | df-sn 4628 | . 2 ⊢ {∅} = {𝑓 ∣ 𝑓 = ∅} | |
6 | 2, 4, 5 | 3eqtr4i 2770 | 1 ⊢ (∅ ↑m ∅) = {∅} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 {cab 2709 ∅c0 4321 {csn 4627 ⟶wf 6536 (class class class)co 7405 ↑m cmap 8816 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-map 8818 |
This theorem is referenced by: efmndbas0 18768 symgvalstruct 19258 symgvalstructOLD 19259 |
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