Theorem fvsingle 35898

Description: The value of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Revised by Scott Fenton, 13-Apr-2018.)

Assertion

Ref	Expression
fvsingle	⊢ (Singleton‘𝐴) = {𝐴}

Proof of Theorem fvsingle

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6822	. . . 4 ⊢ (𝑥 = 𝐴 → (Singleton‘𝑥) = (Singleton‘𝐴))
2		sneq 4587	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
3	1, 2	eqeq12d 2745	. . 3 ⊢ (𝑥 = 𝐴 → ((Singleton‘𝑥) = {𝑥} ↔ (Singleton‘𝐴) = {𝐴}))
4		eqid 2729	. . . . 5 ⊢ {𝑥} = {𝑥}
5		vex 3440	. . . . . 6 ⊢ 𝑥 ∈ V
6		vsnex 5373	. . . . . 6 ⊢ {𝑥} ∈ V
7	5, 6	brsingle 35895	. . . . 5 ⊢ (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥})
8	4, 7	mpbir 231	. . . 4 ⊢ 𝑥Singleton{𝑥}
9		fnsingle 35897	. . . . 5 ⊢ Singleton Fn V
10		fnbrfvb 6873	. . . . 5 ⊢ ((Singleton Fn V ∧ 𝑥 ∈ V) → ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥}))
11	9, 5, 10	mp2an 692	. . . 4 ⊢ ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥})
12	8, 11	mpbir 231	. . 3 ⊢ (Singleton‘𝑥) = {𝑥}
13	3, 12	vtoclg 3509	. 2 ⊢ (𝐴 ∈ V → (Singleton‘𝐴) = {𝐴})
14		fvprc 6814	. . 3 ⊢ (¬ 𝐴 ∈ V → (Singleton‘𝐴) = ∅)
15		snprc 4669	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
16	15	biimpi 216	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
17	14, 16	eqtr4d 2767	. 2 ⊢ (¬ 𝐴 ∈ V → (Singleton‘𝐴) = {𝐴})
18	13, 17	pm2.61i 182	1 ⊢ (Singleton‘𝐴) = {𝐴}

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1540   ∈ wcel 2109  Vcvv 3436  ∅c0 4284  {csn 4577   class class class wbr 5092   Fn wfn 6477  ‘cfv 6482  Singletoncsingle 35816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-symdif 4204  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-eprel 5519  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fo 6488  df-fv 6490  df-1st 7924  df-2nd 7925  df-txp 35832  df-singleton 35840

This theorem is referenced by: (None)