Theorem funressneu 43998

Description: There is exactly one value of a class which is a function restricted to a singleton, analogous to funeu 6361. 𝐴 ∈ V is required because otherwise ∃!𝑦𝐴𝐹𝑦, see brprcneu 6650. (Contributed by AV, 7-Sep-2022.)

Assertion

Ref	Expression
funressneu	⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦)

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

Allowed substitution hints:   𝐵(𝑦)   𝑊(𝑦)

Proof of Theorem funressneu

Step	Hyp	Ref	Expression
1		simp1l 1195	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → 𝐴 ∈ 𝑉)
2		simp1r 1196	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → 𝐵 ∈ 𝑊)
3		simp3 1136	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → 𝐴𝐹𝐵)
4		breldmg 5750	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
5	1, 2, 3, 4	syl3anc 1369	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
6		eldmg 5739	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 → (𝐴 ∈ dom 𝐹 ↔ ∃𝑦 𝐴𝐹𝑦))
7	6	ibi 270	. . 3 ⊢ (𝐴 ∈ dom 𝐹 → ∃𝑦 𝐴𝐹𝑦)
8	5, 7	syl 17	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → ∃𝑦 𝐴𝐹𝑦)
9		simpl 487	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉)
10	9	anim1i 618	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴 ∈ 𝑉 ∧ Fun (𝐹 ↾ {𝐴})))
11	10	3adant3 1130	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → (𝐴 ∈ 𝑉 ∧ Fun (𝐹 ↾ {𝐴})))
12		funressnmo 43997	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ Fun (𝐹 ↾ {𝐴})) → ∃*𝑦 𝐴𝐹𝑦)
13	11, 12	syl 17	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → ∃*𝑦 𝐴𝐹𝑦)
14		moeu 2603	. . 3 ⊢ (∃*𝑦 𝐴𝐹𝑦 ↔ (∃𝑦 𝐴𝐹𝑦 → ∃!𝑦 𝐴𝐹𝑦))
15	13, 14	sylib 221	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → (∃𝑦 𝐴𝐹𝑦 → ∃!𝑦 𝐴𝐹𝑦))
16	8, 15	mpd 15	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1085  ∃wex 1782   ∈ wcel 2112  ∃*wmo 2556  ∃!weu 2588  {csn 4523   class class class wbr 5033  dom cdm 5525   ↾ cres 5527  Fun wfun 6330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-br 5034  df-opab 5096  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-res 5537  df-fun 6338

This theorem is referenced by:  funressnbrafv2  44161