Theorem fsng 5357

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 3409	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
2	1	feq2d 5055	. . 3 ⊢ (𝑎 = 𝐴 → (𝐹:{𝑎}⟶{𝑏} ↔ 𝐹:{𝐴}⟶{𝑏}))
3		opeq1 3570	. . . . 5 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
4	3	sneqd 3411	. . . 4 ⊢ (𝑎 = 𝐴 → {⟨𝑎, 𝑏⟩} = {⟨𝐴, 𝑏⟩})
5	4	eqeq2d 2092	. . 3 ⊢ (𝑎 = 𝐴 → (𝐹 = {⟨𝑎, 𝑏⟩} ↔ 𝐹 = {⟨𝐴, 𝑏⟩}))
6	2, 5	bibi12d 233	. 2 ⊢ (𝑎 = 𝐴 → ((𝐹:{𝑎}⟶{𝑏} ↔ 𝐹 = {⟨𝑎, 𝑏⟩}) ↔ (𝐹:{𝐴}⟶{𝑏} ↔ 𝐹 = {⟨𝐴, 𝑏⟩})))
7		sneq 3409	. . . 4 ⊢ (𝑏 = 𝐵 → {𝑏} = {𝐵})
8		feq3 5052	. . . 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 3571	. . . . 5 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
11	10	sneqd 3411	. . . 4 ⊢ (𝑏 = 𝐵 → {⟨𝐴, 𝑏⟩} = {⟨𝐴, 𝐵⟩})
12	11	eqeq2d 2092	. . 3 ⊢ (𝑏 = 𝐵 → (𝐹 = {⟨𝐴, 𝑏⟩} ↔ 𝐹 = {⟨𝐴, 𝐵⟩}))
13	9, 12	bibi12d 233	. 2 ⊢ (𝑏 = 𝐵 → ((𝐹:{𝐴}⟶{𝑏} ↔ 𝐹 = {⟨𝐴, 𝑏⟩}) ↔ (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩})))
14		vex 2604	. . 3 ⊢ 𝑎 ∈ V
15		vex 2604	. . 3 ⊢ 𝑏 ∈ V
16	14, 15	fsn 5356	. 2 ⊢ (𝐹:{𝑎}⟶{𝑏} ↔ 𝐹 = {⟨𝑎, 𝑏⟩})
17	6, 13, 16	vtocl2g 2662	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩}))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  {csn 3398  ⟨cop 3401  ⟶wf 4918

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929

This theorem is referenced by:  fsn2  5358  xpsng  5359  ftpg  5368  fseq1p1m1  9111