Theorem fsng 7157

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 4641	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
2	1	feq2d 6723	. . 3 ⊢ (𝑎 = 𝐴 → (𝐹:{𝑎}⟶{𝑏} ↔ 𝐹:{𝐴}⟶{𝑏}))
3		opeq1 4878	. . . . 5 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
4	3	sneqd 4643	. . . 4 ⊢ (𝑎 = 𝐴 → {⟨𝑎, 𝑏⟩} = {⟨𝐴, 𝑏⟩})
5	4	eqeq2d 2746	. . 3 ⊢ (𝑎 = 𝐴 → (𝐹 = {⟨𝑎, 𝑏⟩} ↔ 𝐹 = {⟨𝐴, 𝑏⟩}))
6	2, 5	bibi12d 345	. 2 ⊢ (𝑎 = 𝐴 → ((𝐹:{𝑎}⟶{𝑏} ↔ 𝐹 = {⟨𝑎, 𝑏⟩}) ↔ (𝐹:{𝐴}⟶{𝑏} ↔ 𝐹 = {⟨𝐴, 𝑏⟩})))
7		sneq 4641	. . . 4 ⊢ (𝑏 = 𝐵 → {𝑏} = {𝐵})
8	7	feq3d 6724	. . 3 ⊢ (𝑏 = 𝐵 → (𝐹:{𝐴}⟶{𝑏} ↔ 𝐹:{𝐴}⟶{𝐵}))
9		opeq2 4879	. . . . 5 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
10	9	sneqd 4643	. . . 4 ⊢ (𝑏 = 𝐵 → {⟨𝐴, 𝑏⟩} = {⟨𝐴, 𝐵⟩})
11	10	eqeq2d 2746	. . 3 ⊢ (𝑏 = 𝐵 → (𝐹 = {⟨𝐴, 𝑏⟩} ↔ 𝐹 = {⟨𝐴, 𝐵⟩}))
12	8, 11	bibi12d 345	. 2 ⊢ (𝑏 = 𝐵 → ((𝐹:{𝐴}⟶{𝑏} ↔ 𝐹 = {⟨𝐴, 𝑏⟩}) ↔ (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩})))
13		vex 3482	. . 3 ⊢ 𝑎 ∈ V
14		vex 3482	. . 3 ⊢ 𝑏 ∈ V
15	13, 14	fsn 7155	. 2 ⊢ (𝐹:{𝑎}⟶{𝑏} ↔ 𝐹 = {⟨𝑎, 𝑏⟩})
16	6, 12, 15	vtocl2g 3574	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {csn 4631  ⟨cop 4637  ⟶wf 6559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570

This theorem is referenced by:  xpsng  7159  ftpg  7176  mapsnd  8925  axdc3lem4  10491  fseq1p1m1  13635  cats1un  14756  intopsn  18680  efmnd1bas  18919  grp1inv  19079  symg1bas  19423  esumsnf  34045  bnj149  34868  rngosn3  37911  sticksstones9  42136  sticksstones11  42138  k0004lem3  44139  ovnovollem1  46612  mapsnop  48189  snlindsntorlem  48316  lmod1zr  48339