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Theorem fsn2 7155
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 4666 . . . . 5 𝐴 ∈ {𝐴}
3 ffvelcdm 7100 . . . . 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 6089 . . . . . . . . 9 (𝐹 “ dom 𝐹) = ran 𝐹
9 fndm 6671 . . . . . . . . . 10 (𝐹 Fn {𝐴} → dom 𝐹 = {𝐴})
109imaeq2d 6079 . . . . . . . . 9 (𝐹 Fn {𝐴} → (𝐹 “ dom 𝐹) = (𝐹 “ {𝐴}))
118, 10eqtr3id 2788 . . . . . . . 8 (𝐹 Fn {𝐴} → ran 𝐹 = (𝐹 “ {𝐴}))
12 fnsnfv 6987 . . . . . . . . 9 ((𝐹 Fn {𝐴} ∧ 𝐴 ∈ {𝐴}) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
132, 12mpan2 691 . . . . . . . 8 (𝐹 Fn {𝐴} → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
1411, 13eqtr4d 2777 . . . . . . 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 4812 . . . 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 7154 . . 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 1536  wcel 2105  Vcvv 3477  wss 3962  {csn 4630  cop 4636  dom cdm 5688  ran crn 5689  cima 5691   Fn wfn 6557  wf 6558  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570
This theorem is referenced by:  fsn2g  7157  fnressn  7177  fressnfv  7179  mapsnconst  8930  elixpsn  8975  en1  9062  mat1dimelbas  22492  0spth  30154  wlkl0  30395  ldepsnlinclem1  48350  ldepsnlinclem2  48351  0aryfvalel  48483  1arymaptf1  48491
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