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Theorem fvsingle 35921
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 4636 . . . 4 (𝑥 = 𝐴 → {𝑥} = {𝐴})
31, 2eqeq12d 2753 . . 3 (𝑥 = 𝐴 → ((Singleton‘𝑥) = {𝑥} ↔ (Singleton‘𝐴) = {𝐴}))
4 eqid 2737 . . . . 5 {𝑥} = {𝑥}
5 vex 3484 . . . . . 6 𝑥 ∈ V
6 vsnex 5434 . . . . . 6 {𝑥} ∈ V
75, 6brsingle 35918 . . . . 5 (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥})
84, 7mpbir 231 . . . 4 𝑥Singleton{𝑥}
9 fnsingle 35920 . . . . 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 3554 . 2 (𝐴 ∈ V → (Singleton‘𝐴) = {𝐴})
14 fvprc 6898 . . 3 𝐴 ∈ V → (Singleton‘𝐴) = ∅)
15 snprc 4717 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
1615biimpi 216 . . 3 𝐴 ∈ V → {𝐴} = ∅)
1714, 16eqtr4d 2780 . 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 1540  wcel 2108  Vcvv 3480  c0 4333  {csn 4626   class class class wbr 5143   Fn wfn 6556  cfv 6561  Singletoncsingle 35839
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-symdif 4253  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-eprel 5584  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-1st 8014  df-2nd 8015  df-txp 35855  df-singleton 35863
This theorem is referenced by: (None)
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