Theorem fvsingle 36281

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 6871	. . . 4 ⊢ (𝑥 = 𝐴 → (Singleton‘𝑥) = (Singleton‘𝐴))
2		sneq 4595	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
3	1, 2	eqeq12d 2781	. . 3 ⊢ (𝑥 = 𝐴 → ((Singleton‘𝑥) = {𝑥} ↔ (Singleton‘𝐴) = {𝐴}))
4		eqid 2765	. . . . 5 ⊢ {𝑥} = {𝑥}
5		vex 3461	. . . . . 6 ⊢ 𝑥 ∈ V
6		vsnex 5397	. . . . . 6 ⊢ {𝑥} ∈ V
7	5, 6	brsingle 36278	. . . . 5 ⊢ (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥})
8	4, 7	mpbir 234	. . . 4 ⊢ 𝑥Singleton{𝑥}
9		fnsingle 36280	. . . . 5 ⊢ Singleton Fn V
10		fnbrfvb 6921	. . . . 5 ⊢ ((Singleton Fn V ∧ 𝑥 ∈ V) → ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥}))
11	9, 5, 10	mp2an 704	. . . 4 ⊢ ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥})
12	8, 11	mpbir 234	. . 3 ⊢ (Singleton‘𝑥) = {𝑥}
13	3, 12	vtoclg 3525	. 2 ⊢ (𝐴 ∈ V → (Singleton‘𝐴) = {𝐴})
14		fvprc 6863	. . 3 ⊢ (¬ 𝐴 ∈ V → (Singleton‘𝐴) = ∅)
15		snprc 4679	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
16	15	biimpi 219	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
17	14, 16	eqtr4d 2803	. 2 ⊢ (¬ 𝐴 ∈ V → (Singleton‘𝐴) = {𝐴})
18	13, 17	pm2.61i 184	1 ⊢ (Singleton‘𝐴) = {𝐴}

Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1563   ∈ wcel 2145  Vcvv 3457  ∅c0 4288  {csn 4585   class class class wbr 5105   Fn wfn 6520  ‘cfv 6525  Singletoncsingle 36199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-symdif 4208  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-eprel 5552  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fo 6531  df-fv 6533  df-1st 7974  df-2nd 7975  df-txp 36215  df-singleton 36223

This theorem is referenced by: (None)