Theorem 0map0sn0 8924

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.)

Assertion

Ref	Expression
0map0sn0	⊢ (∅ ↑m ∅) = {∅}

Proof of Theorem 0map0sn0

Step	Hyp	Ref	Expression
1		f0bi 6792	. . 3 ⊢ (𝑓:∅⟶∅ ↔ 𝑓 = ∅)
2	1	abbii 2807	. 2 ⊢ {𝑓 ∣ 𝑓:∅⟶∅} = {𝑓 ∣ 𝑓 = ∅}
3		0ex 5313	. . 3 ⊢ ∅ ∈ V
4	3, 3	mapval 8877	. 2 ⊢ (∅ ↑m ∅) = {𝑓 ∣ 𝑓:∅⟶∅}
5		df-sn 4632	. 2 ⊢ {∅} = {𝑓 ∣ 𝑓 = ∅}
6	2, 4, 5	3eqtr4i 2773	1 ⊢ (∅ ↑m ∅) = {∅}

Syntax hints:   = wceq 1537  {cab 2712  ∅c0 4339  {csn 4631  ⟶wf 6559  (class class class)co 7431   ↑_m cmap 8865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867

This theorem is referenced by:  efmndbas0  18917  symgvalstruct  19429  symgvalstructOLD  19430