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Theorem fvsingle 32152
 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.
StepHypRef Expression
1 fveq2 6229 . . . 4 (𝑥 = 𝐴 → (Singleton‘𝑥) = (Singleton‘𝐴))
2 sneq 4220 . . . 4 (𝑥 = 𝐴 → {𝑥} = {𝐴})
31, 2eqeq12d 2666 . . 3 (𝑥 = 𝐴 → ((Singleton‘𝑥) = {𝑥} ↔ (Singleton‘𝐴) = {𝐴}))
4 eqid 2651 . . . . 5 {𝑥} = {𝑥}
5 vex 3234 . . . . . 6 𝑥 ∈ V
6 snex 4938 . . . . . 6 {𝑥} ∈ V
75, 6brsingle 32149 . . . . 5 (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥})
84, 7mpbir 221 . . . 4 𝑥Singleton{𝑥}
9 fnsingle 32151 . . . . 5 Singleton Fn V
10 fnbrfvb 6274 . . . . 5 ((Singleton Fn V ∧ 𝑥 ∈ V) → ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥}))
119, 5, 10mp2an 708 . . . 4 ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥})
128, 11mpbir 221 . . 3 (Singleton‘𝑥) = {𝑥}
133, 12vtoclg 3297 . 2 (𝐴 ∈ V → (Singleton‘𝐴) = {𝐴})
14 fvprc 6223 . . 3 𝐴 ∈ V → (Singleton‘𝐴) = ∅)
15 snprc 4285 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
1615biimpi 206 . . 3 𝐴 ∈ V → {𝐴} = ∅)
1714, 16eqtr4d 2688 . 2 𝐴 ∈ V → (Singleton‘𝐴) = {𝐴})
1813, 17pm2.61i 176 1 (Singleton‘𝐴) = {𝐴}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ∅c0 3948  {csn 4210   class class class wbr 4685   Fn wfn 5921  ‘cfv 5926  Singletoncsingle 32070 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-symdif 3877  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-eprel 5058  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-1st 7210  df-2nd 7211  df-txp 32086  df-singleton 32094 This theorem is referenced by: (None)
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