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Theorem fsn2g 7158
Description: A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by Thierry Arnoux, 11-Jul-2020.)
Assertion
Ref Expression
fsn2g (𝐴𝑉 → (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩})))

Proof of Theorem fsn2g
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 sneq 4636 . . 3 (𝑎 = 𝐴 → {𝑎} = {𝐴})
21feq2d 6722 . 2 (𝑎 = 𝐴 → (𝐹:{𝑎}⟶𝐵𝐹:{𝐴}⟶𝐵))
3 fveq2 6906 . . . 4 (𝑎 = 𝐴 → (𝐹𝑎) = (𝐹𝐴))
43eleq1d 2826 . . 3 (𝑎 = 𝐴 → ((𝐹𝑎) ∈ 𝐵 ↔ (𝐹𝐴) ∈ 𝐵))
5 id 22 . . . . . 6 (𝑎 = 𝐴𝑎 = 𝐴)
65, 3opeq12d 4881 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, (𝐹𝑎)⟩ = ⟨𝐴, (𝐹𝐴)⟩)
76sneqd 4638 . . . 4 (𝑎 = 𝐴 → {⟨𝑎, (𝐹𝑎)⟩} = {⟨𝐴, (𝐹𝐴)⟩})
87eqeq2d 2748 . . 3 (𝑎 = 𝐴 → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))
94, 8anbi12d 632 . 2 (𝑎 = 𝐴 → (((𝐹𝑎) ∈ 𝐵𝐹 = {⟨𝑎, (𝐹𝑎)⟩}) ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩})))
10 vex 3484 . . 3 𝑎 ∈ V
1110fsn2 7156 . 2 (𝐹:{𝑎}⟶𝐵 ↔ ((𝐹𝑎) ∈ 𝐵𝐹 = {⟨𝑎, (𝐹𝑎)⟩}))
122, 9, 11vtoclbg 3557 1 (𝐴𝑉 → (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {csn 4626  cop 4632  wf 6557  cfv 6561
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-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
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-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  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-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by:  fsnex  7303  pt1hmeo  23814  k0004val0  44167  difmapsn  45217  fsetsniunop  47061  f1sn2g  48760
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