Theorem funressnmo 47234

Description: A function restricted to a singleton has at most one value for the singleton element as argument. (Contributed by AV, 2-Sep-2022.)

Assertion

Ref	Expression
funressnmo	⊢ ((𝐴 ∈ 𝑉 ∧ Fun (𝐹 ↾ {𝐴})) → ∃*𝑦 𝐴𝐹𝑦)

Distinct variable groups:   𝑦,𝐴   𝑦,𝐹   𝑦,𝑉

Proof of Theorem funressnmo

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4588	. . . . . 6 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	reseq2d 5936	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐴}))
3	2	funeqd 6512	. . . 4 ⊢ (𝑥 = 𝐴 → (Fun (𝐹 ↾ {𝑥}) ↔ Fun (𝐹 ↾ {𝐴})))
4		breq1 5099	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
5	4	mobidv 2547	. . . 4 ⊢ (𝑥 = 𝐴 → (∃*𝑦 𝑥𝐹𝑦 ↔ ∃*𝑦 𝐴𝐹𝑦))
6	3, 5	imbi12d 344	. . 3 ⊢ (𝑥 = 𝐴 → ((Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦) ↔ (Fun (𝐹 ↾ {𝐴}) → ∃*𝑦 𝐴𝐹𝑦)))
7		funressnvmo 47233	. . 3 ⊢ (Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦)
8	6, 7	vtoclg 3509	. 2 ⊢ (𝐴 ∈ 𝑉 → (Fun (𝐹 ↾ {𝐴}) → ∃*𝑦 𝐴𝐹𝑦))
9	8	imp 406	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Fun (𝐹 ↾ {𝐴})) → ∃*𝑦 𝐴𝐹𝑦)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃*wmo 2535  {csn 4578   class class class wbr 5096   ↾ cres 5624  Fun wfun 6484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2537  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-res 5634  df-fun 6492

This theorem is referenced by:  funressneu  47235