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Theorem fsn2 6363
 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 4184 . . . . 5 𝐴 ∈ {𝐴}
3 ffvelrn 6318 . . . . 5 ((𝐹:{𝐴}⟶𝐵𝐴 ∈ {𝐴}) → (𝐹𝐴) ∈ 𝐵)
42, 3mpan2 706 . . . 4 (𝐹:{𝐴}⟶𝐵 → (𝐹𝐴) ∈ 𝐵)
5 ffn 6007 . . . . 5 (𝐹:{𝐴}⟶𝐵𝐹 Fn {𝐴})
6 dffn3 6016 . . . . . . 7 (𝐹 Fn {𝐴} ↔ 𝐹:{𝐴}⟶ran 𝐹)
76biimpi 206 . . . . . 6 (𝐹 Fn {𝐴} → 𝐹:{𝐴}⟶ran 𝐹)
8 imadmrn 5440 . . . . . . . . 9 (𝐹 “ dom 𝐹) = ran 𝐹
9 fndm 5953 . . . . . . . . . 10 (𝐹 Fn {𝐴} → dom 𝐹 = {𝐴})
109imaeq2d 5430 . . . . . . . . 9 (𝐹 Fn {𝐴} → (𝐹 “ dom 𝐹) = (𝐹 “ {𝐴}))
118, 10syl5eqr 2669 . . . . . . . 8 (𝐹 Fn {𝐴} → ran 𝐹 = (𝐹 “ {𝐴}))
12 fnsnfv 6220 . . . . . . . . 9 ((𝐹 Fn {𝐴} ∧ 𝐴 ∈ {𝐴}) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
132, 12mpan2 706 . . . . . . . 8 (𝐹 Fn {𝐴} → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
1411, 13eqtr4d 2658 . . . . . . 7 (𝐹 Fn {𝐴} → ran 𝐹 = {(𝐹𝐴)})
1514feq3d 5994 . . . . . 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 554 . . 3 (𝐹:{𝐴}⟶𝐵 → ((𝐹𝐴) ∈ 𝐵𝐹:{𝐴}⟶{(𝐹𝐴)}))
19 snssi 4313 . . . 4 ((𝐹𝐴) ∈ 𝐵 → {(𝐹𝐴)} ⊆ 𝐵)
20 fss 6018 . . . . 5 ((𝐹:{𝐴}⟶{(𝐹𝐴)} ∧ {(𝐹𝐴)} ⊆ 𝐵) → 𝐹:{𝐴}⟶𝐵)
2120ancoms 469 . . . 4 (({(𝐹𝐴)} ⊆ 𝐵𝐹:{𝐴}⟶{(𝐹𝐴)}) → 𝐹:{𝐴}⟶𝐵)
2219, 21sylan 488 . . 3 (((𝐹𝐴) ∈ 𝐵𝐹:{𝐴}⟶{(𝐹𝐴)}) → 𝐹:{𝐴}⟶𝐵)
2318, 22impbii 199 . 2 (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹:{𝐴}⟶{(𝐹𝐴)}))
24 fvex 6163 . . . 4 (𝐹𝐴) ∈ V
251, 24fsn 6362 . . 3 (𝐹:{𝐴}⟶{(𝐹𝐴)} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
2625anbi2i 729 . 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 384   = wceq 1480   ∈ wcel 1987  Vcvv 3189   ⊆ wss 3559  {csn 4153  ⟨cop 4159  dom cdm 5079  ran crn 5080   “ cima 5082   Fn wfn 5847  ⟶wf 5848  ‘cfv 5852 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860 This theorem is referenced by:  fsn2g  6365  fnressn  6385  fressnfv  6387  mapsnconst  7855  elixpsn  7899  en1  7975  mat1dimelbas  20209  0spth  26870  ldepsnlinclem1  41608  ldepsnlinclem2  41609
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