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Theorem fsn2 6879
 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 4564 . . . . 5 𝐴 ∈ {𝐴}
3 ffvelrn 6830 . . . . 5 ((𝐹:{𝐴}⟶𝐵𝐴 ∈ {𝐴}) → (𝐹𝐴) ∈ 𝐵)
42, 3mpan2 690 . . . 4 (𝐹:{𝐴}⟶𝐵 → (𝐹𝐴) ∈ 𝐵)
5 ffn 6491 . . . . 5 (𝐹:{𝐴}⟶𝐵𝐹 Fn {𝐴})
6 dffn3 6503 . . . . . . 7 (𝐹 Fn {𝐴} ↔ 𝐹:{𝐴}⟶ran 𝐹)
76biimpi 219 . . . . . 6 (𝐹 Fn {𝐴} → 𝐹:{𝐴}⟶ran 𝐹)
8 imadmrn 5910 . . . . . . . . 9 (𝐹 “ dom 𝐹) = ran 𝐹
9 fndm 6429 . . . . . . . . . 10 (𝐹 Fn {𝐴} → dom 𝐹 = {𝐴})
109imaeq2d 5900 . . . . . . . . 9 (𝐹 Fn {𝐴} → (𝐹 “ dom 𝐹) = (𝐹 “ {𝐴}))
118, 10syl5eqr 2850 . . . . . . . 8 (𝐹 Fn {𝐴} → ran 𝐹 = (𝐹 “ {𝐴}))
12 fnsnfv 6722 . . . . . . . . 9 ((𝐹 Fn {𝐴} ∧ 𝐴 ∈ {𝐴}) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
132, 12mpan2 690 . . . . . . . 8 (𝐹 Fn {𝐴} → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
1411, 13eqtr4d 2839 . . . . . . 7 (𝐹 Fn {𝐴} → ran 𝐹 = {(𝐹𝐴)})
1514feq3d 6478 . . . . . 6 (𝐹 Fn {𝐴} → (𝐹:{𝐴}⟶ran 𝐹𝐹:{𝐴}⟶{(𝐹𝐴)}))
167, 15mpbid 235 . . . . 5 (𝐹 Fn {𝐴} → 𝐹:{𝐴}⟶{(𝐹𝐴)})
175, 16syl 17 . . . 4 (𝐹:{𝐴}⟶𝐵𝐹:{𝐴}⟶{(𝐹𝐴)})
184, 17jca 515 . . 3 (𝐹:{𝐴}⟶𝐵 → ((𝐹𝐴) ∈ 𝐵𝐹:{𝐴}⟶{(𝐹𝐴)}))
19 snssi 4704 . . . 4 ((𝐹𝐴) ∈ 𝐵 → {(𝐹𝐴)} ⊆ 𝐵)
20 fss 6505 . . . . 5 ((𝐹:{𝐴}⟶{(𝐹𝐴)} ∧ {(𝐹𝐴)} ⊆ 𝐵) → 𝐹:{𝐴}⟶𝐵)
2120ancoms 462 . . . 4 (({(𝐹𝐴)} ⊆ 𝐵𝐹:{𝐴}⟶{(𝐹𝐴)}) → 𝐹:{𝐴}⟶𝐵)
2219, 21sylan 583 . . 3 (((𝐹𝐴) ∈ 𝐵𝐹:{𝐴}⟶{(𝐹𝐴)}) → 𝐹:{𝐴}⟶𝐵)
2318, 22impbii 212 . 2 (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹:{𝐴}⟶{(𝐹𝐴)}))
24 fvex 6662 . . . 4 (𝐹𝐴) ∈ V
251, 24fsn 6878 . . 3 (𝐹:{𝐴}⟶{(𝐹𝐴)} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
2625anbi2i 625 . 2 (((𝐹𝐴) ∈ 𝐵𝐹:{𝐴}⟶{(𝐹𝐴)}) ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))
2723, 26bitri 278 1 (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2112  Vcvv 3444   ⊆ wss 3884  {csn 4528  ⟨cop 4534  dom cdm 5523  ran crn 5524   “ cima 5526   Fn wfn 6323  ⟶wf 6324  ‘cfv 6328 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336 This theorem is referenced by:  fsn2g  6881  fnressn  6901  fressnfv  6903  mapsnconst  8443  elixpsn  8488  en1  8563  mat1dimelbas  21080  0spth  27915  wlkl0  28156  ldepsnlinclem1  44911  ldepsnlinclem2  44912  0aryfvalel  45045  1arymaptf1  45053
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