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Theorem fvsingle 35901
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 6906 . . . 4 (𝑥 = 𝐴 → (Singleton‘𝑥) = (Singleton‘𝐴))
2 sneq 4640 . . . 4 (𝑥 = 𝐴 → {𝑥} = {𝐴})
31, 2eqeq12d 2750 . . 3 (𝑥 = 𝐴 → ((Singleton‘𝑥) = {𝑥} ↔ (Singleton‘𝐴) = {𝐴}))
4 eqid 2734 . . . . 5 {𝑥} = {𝑥}
5 vex 3481 . . . . . 6 𝑥 ∈ V
6 vsnex 5439 . . . . . 6 {𝑥} ∈ V
75, 6brsingle 35898 . . . . 5 (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥})
84, 7mpbir 231 . . . 4 𝑥Singleton{𝑥}
9 fnsingle 35900 . . . . 5 Singleton Fn V
10 fnbrfvb 6959 . . . . 5 ((Singleton Fn V ∧ 𝑥 ∈ V) → ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥}))
119, 5, 10mp2an 692 . . . 4 ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥})
128, 11mpbir 231 . . 3 (Singleton‘𝑥) = {𝑥}
133, 12vtoclg 3553 . 2 (𝐴 ∈ V → (Singleton‘𝐴) = {𝐴})
14 fvprc 6898 . . 3 𝐴 ∈ V → (Singleton‘𝐴) = ∅)
15 snprc 4721 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
1615biimpi 216 . . 3 𝐴 ∈ V → {𝐴} = ∅)
1714, 16eqtr4d 2777 . 2 𝐴 ∈ V → (Singleton‘𝐴) = {𝐴})
1813, 17pm2.61i 182 1 (Singleton‘𝐴) = {𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1536  wcel 2105  Vcvv 3477  c0 4338  {csn 4630   class class class wbr 5147   Fn wfn 6557  cfv 6562  Singletoncsingle 35819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-symdif 4258  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-eprel 5588  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fo 6568  df-fv 6570  df-1st 8012  df-2nd 8013  df-txp 35835  df-singleton 35843
This theorem is referenced by: (None)
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