Theorem funressnmo 43280

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 4576	. . . . . 6 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	reseq2d 5852	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐴}))
3	2	funeqd 6376	. . . 4 ⊢ (𝑥 = 𝐴 → (Fun (𝐹 ↾ {𝑥}) ↔ Fun (𝐹 ↾ {𝐴})))
4		breq1 5068	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
5	4	mobidv 2629	. . . 4 ⊢ (𝑥 = 𝐴 → (∃*𝑦 𝑥𝐹𝑦 ↔ ∃*𝑦 𝐴𝐹𝑦))
6	3, 5	imbi12d 347	. . 3 ⊢ (𝑥 = 𝐴 → ((Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦) ↔ (Fun (𝐹 ↾ {𝐴}) → ∃*𝑦 𝐴𝐹𝑦)))
7		funressnvmo 43279	. . 3 ⊢ (Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦)
8	6, 7	vtoclg 3567	. 2 ⊢ (𝐴 ∈ 𝑉 → (Fun (𝐹 ↾ {𝐴}) → ∃*𝑦 𝐴𝐹𝑦))
9	8	imp 409	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Fun (𝐹 ↾ {𝐴})) → ∃*𝑦 𝐴𝐹𝑦)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∃*wmo 2616  {csn 4566   class class class wbr 5065   ↾ cres 5556  Fun wfun 6348

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-res 5566  df-fun 6356

This theorem is referenced by:  funressneu  43281