Theorem fsn2g 7072

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	⊢ (𝐴 ∈ 𝑉 → (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})))

Proof of Theorem fsn2g

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4587	. . 3 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
2	1	feq2d 6636	. 2 ⊢ (𝑎 = 𝐴 → (𝐹:{𝑎}⟶𝐵 ↔ 𝐹:{𝐴}⟶𝐵))
3		fveq2 6822	. . . 4 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴))
4	3	eleq1d 2813	. . 3 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑎) ∈ 𝐵 ↔ (𝐹‘𝐴) ∈ 𝐵))
5		id 22	. . . . . 6 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
6	5, 3	opeq12d 4832	. . . . 5 ⊢ (𝑎 = 𝐴 → ⟨𝑎, (𝐹‘𝑎)⟩ = ⟨𝐴, (𝐹‘𝐴)⟩)
7	6	sneqd 4589	. . . 4 ⊢ (𝑎 = 𝐴 → {⟨𝑎, (𝐹‘𝑎)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩})
8	7	eqeq2d 2740	. . 3 ⊢ (𝑎 = 𝐴 → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}))
9	4, 8	anbi12d 632	. 2 ⊢ (𝑎 = 𝐴 → (((𝐹‘𝑎) ∈ 𝐵 ∧ 𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩}) ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})))
10		vex 3440	. . 3 ⊢ 𝑎 ∈ V
11	10	fsn2 7070	. 2 ⊢ (𝐹:{𝑎}⟶𝐵 ↔ ((𝐹‘𝑎) ∈ 𝐵 ∧ 𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩}))
12	2, 9, 11	vtoclbg 3512	1 ⊢ (𝐴 ∈ 𝑉 → (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {csn 4577  ⟨cop 4583  ⟶wf 6478  ‘cfv 6482

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490

This theorem is referenced by:  fsnex  7220  pt1hmeo  23691  k0004val0  44127  difmapsn  45190  fsetsniunop  47033  f1sn2g  48835  termcfuncval  49517