Theorem fsn2g 6877

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 4535	. . 3 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
2	1	feq2d 6473	. 2 ⊢ (𝑎 = 𝐴 → (𝐹:{𝑎}⟶𝐵 ↔ 𝐹:{𝐴}⟶𝐵))
3		fveq2 6645	. . . 4 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴))
4	3	eleq1d 2874	. . 3 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑎) ∈ 𝐵 ↔ (𝐹‘𝐴) ∈ 𝐵))
5		id 22	. . . . . 6 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
6	5, 3	opeq12d 4773	. . . . 5 ⊢ (𝑎 = 𝐴 → ⟨𝑎, (𝐹‘𝑎)⟩ = ⟨𝐴, (𝐹‘𝐴)⟩)
7	6	sneqd 4537	. . . 4 ⊢ (𝑎 = 𝐴 → {⟨𝑎, (𝐹‘𝑎)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩})
8	7	eqeq2d 2809	. . 3 ⊢ (𝑎 = 𝐴 → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}))
9	4, 8	anbi12d 633	. 2 ⊢ (𝑎 = 𝐴 → (((𝐹‘𝑎) ∈ 𝐵 ∧ 𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩}) ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})))
10		vex 3444	. . 3 ⊢ 𝑎 ∈ V
11	10	fsn2 6875	. 2 ⊢ (𝐹:{𝑎}⟶𝐵 ↔ ((𝐹‘𝑎) ∈ 𝐵 ∧ 𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩}))
12	2, 9, 11	vtoclbg 3517	1 ⊢ (𝐴 ∈ 𝑉 → (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {csn 4525  ⟨cop 4531  ⟶wf 6320  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332

This theorem is referenced by:  fsnex  7017  pt1hmeo  22411  k0004val0  40857  difmapsn  41841