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Theorem fsn2g 6362
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 4160 . . . 4 (𝑎 = 𝐴 → {𝑎} = {𝐴})
21feq2d 5990 . . 3 (𝑎 = 𝐴 → (𝐹:{𝑎}⟶𝐵𝐹:{𝐴}⟶𝐵))
3 fveq2 6150 . . . . 5 (𝑎 = 𝐴 → (𝐹𝑎) = (𝐹𝐴))
43eleq1d 2683 . . . 4 (𝑎 = 𝐴 → ((𝐹𝑎) ∈ 𝐵 ↔ (𝐹𝐴) ∈ 𝐵))
5 eqidd 2622 . . . . 5 (𝑎 = 𝐴𝐹 = 𝐹)
6 id 22 . . . . . . 7 (𝑎 = 𝐴𝑎 = 𝐴)
76, 3opeq12d 4380 . . . . . 6 (𝑎 = 𝐴 → ⟨𝑎, (𝐹𝑎)⟩ = ⟨𝐴, (𝐹𝐴)⟩)
87sneqd 4162 . . . . 5 (𝑎 = 𝐴 → {⟨𝑎, (𝐹𝑎)⟩} = {⟨𝐴, (𝐹𝐴)⟩})
95, 8eqeq12d 2636 . . . 4 (𝑎 = 𝐴 → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))
104, 9anbi12d 746 . . 3 (𝑎 = 𝐴 → (((𝐹𝑎) ∈ 𝐵𝐹 = {⟨𝑎, (𝐹𝑎)⟩}) ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩})))
112, 10bibi12d 335 . 2 (𝑎 = 𝐴 → ((𝐹:{𝑎}⟶𝐵 ↔ ((𝐹𝑎) ∈ 𝐵𝐹 = {⟨𝑎, (𝐹𝑎)⟩})) ↔ (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))))
12 vex 3189 . . 3 𝑎 ∈ V
1312fsn2 6360 . 2 (𝐹:{𝑎}⟶𝐵 ↔ ((𝐹𝑎) ∈ 𝐵𝐹 = {⟨𝑎, (𝐹𝑎)⟩}))
1411, 13vtoclg 3252 1 (𝐴𝑉 → (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {csn 4150  cop 4156  wf 5845  cfv 5849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857
This theorem is referenced by:  fsnex  6495  pt1hmeo  21522  k0004val0  37955  difmapsn  38896
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