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Theorem fsn2 6443
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 4241 . . . . 5 𝐴 ∈ {𝐴}
3 ffvelrn 6397 . . . . 5 ((𝐹:{𝐴}⟶𝐵𝐴 ∈ {𝐴}) → (𝐹𝐴) ∈ 𝐵)
42, 3mpan2 707 . . . 4 (𝐹:{𝐴}⟶𝐵 → (𝐹𝐴) ∈ 𝐵)
5 ffn 6083 . . . . 5 (𝐹:{𝐴}⟶𝐵𝐹 Fn {𝐴})
6 dffn3 6092 . . . . . . 7 (𝐹 Fn {𝐴} ↔ 𝐹:{𝐴}⟶ran 𝐹)
76biimpi 206 . . . . . 6 (𝐹 Fn {𝐴} → 𝐹:{𝐴}⟶ran 𝐹)
8 imadmrn 5511 . . . . . . . . 9 (𝐹 “ dom 𝐹) = ran 𝐹
9 fndm 6028 . . . . . . . . . 10 (𝐹 Fn {𝐴} → dom 𝐹 = {𝐴})
109imaeq2d 5501 . . . . . . . . 9 (𝐹 Fn {𝐴} → (𝐹 “ dom 𝐹) = (𝐹 “ {𝐴}))
118, 10syl5eqr 2699 . . . . . . . 8 (𝐹 Fn {𝐴} → ran 𝐹 = (𝐹 “ {𝐴}))
12 fnsnfv 6297 . . . . . . . . 9 ((𝐹 Fn {𝐴} ∧ 𝐴 ∈ {𝐴}) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
132, 12mpan2 707 . . . . . . . 8 (𝐹 Fn {𝐴} → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
1411, 13eqtr4d 2688 . . . . . . 7 (𝐹 Fn {𝐴} → ran 𝐹 = {(𝐹𝐴)})
1514feq3d 6070 . . . . . 6 (𝐹 Fn {𝐴} → (𝐹:{𝐴}⟶ran 𝐹𝐹:{𝐴}⟶{(𝐹𝐴)}))
167, 15mpbid 222 . . . . 5 (𝐹 Fn {𝐴} → 𝐹:{𝐴}⟶{(𝐹𝐴)})
175, 16syl 17 . . . 4 (𝐹:{𝐴}⟶𝐵𝐹:{𝐴}⟶{(𝐹𝐴)})
184, 17jca 553 . . 3 (𝐹:{𝐴}⟶𝐵 → ((𝐹𝐴) ∈ 𝐵𝐹:{𝐴}⟶{(𝐹𝐴)}))
19 snssi 4371 . . . 4 ((𝐹𝐴) ∈ 𝐵 → {(𝐹𝐴)} ⊆ 𝐵)
20 fss 6094 . . . . 5 ((𝐹:{𝐴}⟶{(𝐹𝐴)} ∧ {(𝐹𝐴)} ⊆ 𝐵) → 𝐹:{𝐴}⟶𝐵)
2120ancoms 468 . . . 4 (({(𝐹𝐴)} ⊆ 𝐵𝐹:{𝐴}⟶{(𝐹𝐴)}) → 𝐹:{𝐴}⟶𝐵)
2219, 21sylan 487 . . 3 (((𝐹𝐴) ∈ 𝐵𝐹:{𝐴}⟶{(𝐹𝐴)}) → 𝐹:{𝐴}⟶𝐵)
2318, 22impbii 199 . 2 (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹:{𝐴}⟶{(𝐹𝐴)}))
24 fvex 6239 . . . 4 (𝐹𝐴) ∈ V
251, 24fsn 6442 . . 3 (𝐹:{𝐴}⟶{(𝐹𝐴)} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
2625anbi2i 730 . 2 (((𝐹𝐴) ∈ 𝐵𝐹:{𝐴}⟶{(𝐹𝐴)}) ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))
2723, 26bitri 264 1 (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  wss 3607  {csn 4210  cop 4216  dom cdm 5143  ran crn 5144  cima 5146   Fn wfn 5921  wf 5922  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934
This theorem is referenced by:  fsn2g  6445  fnressn  6465  fressnfv  6467  mapsnconst  7945  elixpsn  7989  en1  8064  mat1dimelbas  20325  0spth  27104  ldepsnlinclem1  42619  ldepsnlinclem2  42620
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