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Theorem fsn2 7156
Description: A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.)
Hypothesis
Ref Expression
fsn2.1 𝐴 ∈ V
Assertion
Ref Expression
fsn2 (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))

Proof of Theorem fsn2
StepHypRef Expression
1 fsn2.1 . . . . . 6 𝐴 ∈ V
21snid 4662 . . . . 5 𝐴 ∈ {𝐴}
3 ffvelcdm 7101 . . . . 5 ((𝐹:{𝐴}⟶𝐵𝐴 ∈ {𝐴}) → (𝐹𝐴) ∈ 𝐵)
42, 3mpan2 691 . . . 4 (𝐹:{𝐴}⟶𝐵 → (𝐹𝐴) ∈ 𝐵)
5 ffn 6736 . . . . 5 (𝐹:{𝐴}⟶𝐵𝐹 Fn {𝐴})
6 dffn3 6748 . . . . . . 7 (𝐹 Fn {𝐴} ↔ 𝐹:{𝐴}⟶ran 𝐹)
76biimpi 216 . . . . . 6 (𝐹 Fn {𝐴} → 𝐹:{𝐴}⟶ran 𝐹)
8 imadmrn 6088 . . . . . . . . 9 (𝐹 “ dom 𝐹) = ran 𝐹
9 fndm 6671 . . . . . . . . . 10 (𝐹 Fn {𝐴} → dom 𝐹 = {𝐴})
109imaeq2d 6078 . . . . . . . . 9 (𝐹 Fn {𝐴} → (𝐹 “ dom 𝐹) = (𝐹 “ {𝐴}))
118, 10eqtr3id 2791 . . . . . . . 8 (𝐹 Fn {𝐴} → ran 𝐹 = (𝐹 “ {𝐴}))
12 fnsnfv 6988 . . . . . . . . 9 ((𝐹 Fn {𝐴} ∧ 𝐴 ∈ {𝐴}) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
132, 12mpan2 691 . . . . . . . 8 (𝐹 Fn {𝐴} → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
1411, 13eqtr4d 2780 . . . . . . 7 (𝐹 Fn {𝐴} → ran 𝐹 = {(𝐹𝐴)})
1514feq3d 6723 . . . . . 6 (𝐹 Fn {𝐴} → (𝐹:{𝐴}⟶ran 𝐹𝐹:{𝐴}⟶{(𝐹𝐴)}))
167, 15mpbid 232 . . . . 5 (𝐹 Fn {𝐴} → 𝐹:{𝐴}⟶{(𝐹𝐴)})
175, 16syl 17 . . . 4 (𝐹:{𝐴}⟶𝐵𝐹:{𝐴}⟶{(𝐹𝐴)})
184, 17jca 511 . . 3 (𝐹:{𝐴}⟶𝐵 → ((𝐹𝐴) ∈ 𝐵𝐹:{𝐴}⟶{(𝐹𝐴)}))
19 snssi 4808 . . . 4 ((𝐹𝐴) ∈ 𝐵 → {(𝐹𝐴)} ⊆ 𝐵)
20 fss 6752 . . . . 5 ((𝐹:{𝐴}⟶{(𝐹𝐴)} ∧ {(𝐹𝐴)} ⊆ 𝐵) → 𝐹:{𝐴}⟶𝐵)
2120ancoms 458 . . . 4 (({(𝐹𝐴)} ⊆ 𝐵𝐹:{𝐴}⟶{(𝐹𝐴)}) → 𝐹:{𝐴}⟶𝐵)
2219, 21sylan 580 . . 3 (((𝐹𝐴) ∈ 𝐵𝐹:{𝐴}⟶{(𝐹𝐴)}) → 𝐹:{𝐴}⟶𝐵)
2318, 22impbii 209 . 2 (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹:{𝐴}⟶{(𝐹𝐴)}))
24 fvex 6919 . . . 4 (𝐹𝐴) ∈ V
251, 24fsn 7155 . . 3 (𝐹:{𝐴}⟶{(𝐹𝐴)} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
2625anbi2i 623 . 2 (((𝐹𝐴) ∈ 𝐵𝐹:{𝐴}⟶{(𝐹𝐴)}) ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))
2723, 26bitri 275 1 (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  wss 3951  {csn 4626  cop 4632  dom cdm 5685  ran crn 5686  cima 5688   Fn wfn 6556  wf 6557  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by:  fsn2g  7158  fnressn  7178  fressnfv  7180  mapsnconst  8932  elixpsn  8977  en1  9064  mat1dimelbas  22477  0spth  30145  wlkl0  30386  ldepsnlinclem1  48422  ldepsnlinclem2  48423  0aryfvalel  48555  1arymaptf1  48563
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