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Theorem fsn2g 6548
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 4326 . . . 4 (𝑎 = 𝐴 → {𝑎} = {𝐴})
21feq2d 6171 . . 3 (𝑎 = 𝐴 → (𝐹:{𝑎}⟶𝐵𝐹:{𝐴}⟶𝐵))
3 fveq2 6332 . . . . 5 (𝑎 = 𝐴 → (𝐹𝑎) = (𝐹𝐴))
43eleq1d 2835 . . . 4 (𝑎 = 𝐴 → ((𝐹𝑎) ∈ 𝐵 ↔ (𝐹𝐴) ∈ 𝐵))
5 eqidd 2772 . . . . 5 (𝑎 = 𝐴𝐹 = 𝐹)
6 id 22 . . . . . . 7 (𝑎 = 𝐴𝑎 = 𝐴)
76, 3opeq12d 4547 . . . . . 6 (𝑎 = 𝐴 → ⟨𝑎, (𝐹𝑎)⟩ = ⟨𝐴, (𝐹𝐴)⟩)
87sneqd 4328 . . . . 5 (𝑎 = 𝐴 → {⟨𝑎, (𝐹𝑎)⟩} = {⟨𝐴, (𝐹𝐴)⟩})
95, 8eqeq12d 2786 . . . 4 (𝑎 = 𝐴 → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))
104, 9anbi12d 616 . . 3 (𝑎 = 𝐴 → (((𝐹𝑎) ∈ 𝐵𝐹 = {⟨𝑎, (𝐹𝑎)⟩}) ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩})))
112, 10bibi12d 334 . 2 (𝑎 = 𝐴 → ((𝐹:{𝑎}⟶𝐵 ↔ ((𝐹𝑎) ∈ 𝐵𝐹 = {⟨𝑎, (𝐹𝑎)⟩})) ↔ (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))))
12 vex 3354 . . 3 𝑎 ∈ V
1312fsn2 6546 . 2 (𝐹:{𝑎}⟶𝐵 ↔ ((𝐹𝑎) ∈ 𝐵𝐹 = {⟨𝑎, (𝐹𝑎)⟩}))
1411, 13vtoclg 3417 1 (𝐴𝑉 → (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  {csn 4316  cop 4322  wf 6027  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039
This theorem is referenced by:  fsnex  6681  pt1hmeo  21830  k0004val0  38978  difmapsn  39922
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