Theorem fsng 5820

Description: A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.)

Assertion

Ref	Expression
fsng	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩}))

Proof of Theorem fsng

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 3680	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
2	1	feq2d 5470	. . 3 ⊢ (𝑎 = 𝐴 → (𝐹:{𝑎}⟶{𝑏} ↔ 𝐹:{𝐴}⟶{𝑏}))
3		opeq1 3862	. . . . 5 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
4	3	sneqd 3682	. . . 4 ⊢ (𝑎 = 𝐴 → {⟨𝑎, 𝑏⟩} = {⟨𝐴, 𝑏⟩})
5	4	eqeq2d 2243	. . 3 ⊢ (𝑎 = 𝐴 → (𝐹 = {⟨𝑎, 𝑏⟩} ↔ 𝐹 = {⟨𝐴, 𝑏⟩}))
6	2, 5	bibi12d 235	. 2 ⊢ (𝑎 = 𝐴 → ((𝐹:{𝑎}⟶{𝑏} ↔ 𝐹 = {⟨𝑎, 𝑏⟩}) ↔ (𝐹:{𝐴}⟶{𝑏} ↔ 𝐹 = {⟨𝐴, 𝑏⟩})))
7		sneq 3680	. . . 4 ⊢ (𝑏 = 𝐵 → {𝑏} = {𝐵})
8		feq3 5467	. . . 4 ⊢ ({𝑏} = {𝐵} → (𝐹:{𝐴}⟶{𝑏} ↔ 𝐹:{𝐴}⟶{𝐵}))
9	7, 8	syl 14	. . 3 ⊢ (𝑏 = 𝐵 → (𝐹:{𝐴}⟶{𝑏} ↔ 𝐹:{𝐴}⟶{𝐵}))
10		opeq2 3863	. . . . 5 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
11	10	sneqd 3682	. . . 4 ⊢ (𝑏 = 𝐵 → {⟨𝐴, 𝑏⟩} = {⟨𝐴, 𝐵⟩})
12	11	eqeq2d 2243	. . 3 ⊢ (𝑏 = 𝐵 → (𝐹 = {⟨𝐴, 𝑏⟩} ↔ 𝐹 = {⟨𝐴, 𝐵⟩}))
13	9, 12	bibi12d 235	. 2 ⊢ (𝑏 = 𝐵 → ((𝐹:{𝐴}⟶{𝑏} ↔ 𝐹 = {⟨𝐴, 𝑏⟩}) ↔ (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩})))
14		vex 2805	. . 3 ⊢ 𝑎 ∈ V
15		vex 2805	. . 3 ⊢ 𝑏 ∈ V
16	14, 15	fsn 5819	. 2 ⊢ (𝐹:{𝑎}⟶{𝑏} ↔ 𝐹 = {⟨𝑎, 𝑏⟩})
17	6, 13, 16	vtocl2g 2868	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩}))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202  {csn 3669  ⟨cop 3672  ⟶wf 5322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333

This theorem is referenced by:  fsn2  5821  xpsng  5822  ftpg  5837  fseq1p1m1  10328  cats1un  11301  intopsn  13449  grp1inv  13689