Theorem fsn2g 7081

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 4588	. . 3 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
2	1	feq2d 6644	. 2 ⊢ (𝑎 = 𝐴 → (𝐹:{𝑎}⟶𝐵 ↔ 𝐹:{𝐴}⟶𝐵))
3		fveq2 6832	. . . 4 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴))
4	3	eleq1d 2819	. . 3 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑎) ∈ 𝐵 ↔ (𝐹‘𝐴) ∈ 𝐵))
5		id 22	. . . . . 6 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
6	5, 3	opeq12d 4835	. . . . 5 ⊢ (𝑎 = 𝐴 → ⟨𝑎, (𝐹‘𝑎)⟩ = ⟨𝐴, (𝐹‘𝐴)⟩)
7	6	sneqd 4590	. . . 4 ⊢ (𝑎 = 𝐴 → {⟨𝑎, (𝐹‘𝑎)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩})
8	7	eqeq2d 2745	. . 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 3442	. . 3 ⊢ 𝑎 ∈ V
11	10	fsn2 7079	. 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 1541   ∈ wcel 2113  {csn 4578  ⟨cop 4584  ⟶wf 6486  ‘cfv 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498

This theorem is referenced by:  fsnex  7227  pt1hmeo  23748  k0004val0  44337  difmapsn  45398  fsetsniunop  47237  f1sn2g  49038  termcfuncval  49719