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Theorem fsn2 5262
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) B 𝐹 = {⟨A, (𝐹A)⟩}))

Proof of Theorem fsn2
StepHypRef Expression
1 ffn 4972 . . 3 (𝐹:{A}⟶B𝐹 Fn {A})
2 fsn2.1 . . . . 5 A V
32snid 3377 . . . 4 A {A}
4 funfvex 5117 . . . . 5 ((Fun 𝐹 A dom 𝐹) → (𝐹A) V)
54funfni 4925 . . . 4 ((𝐹 Fn {A} A {A}) → (𝐹A) V)
63, 5mpan2 403 . . 3 (𝐹 Fn {A} → (𝐹A) V)
71, 6syl 14 . 2 (𝐹:{A}⟶B → (𝐹A) V)
8 elex 2543 . . 3 ((𝐹A) B → (𝐹A) V)
98adantr 261 . 2 (((𝐹A) B 𝐹 = {⟨A, (𝐹A)⟩}) → (𝐹A) V)
10 ffvelrn 5225 . . . . . 6 ((𝐹:{A}⟶B A {A}) → (𝐹A) B)
113, 10mpan2 403 . . . . 5 (𝐹:{A}⟶B → (𝐹A) B)
12 dffn3 4979 . . . . . . . 8 (𝐹 Fn {A} ↔ 𝐹:{A}⟶ran 𝐹)
1312biimpi 113 . . . . . . 7 (𝐹 Fn {A} → 𝐹:{A}⟶ran 𝐹)
14 imadmrn 4605 . . . . . . . . . 10 (𝐹 “ dom 𝐹) = ran 𝐹
15 fndm 4924 . . . . . . . . . . 11 (𝐹 Fn {A} → dom 𝐹 = {A})
1615imaeq2d 4595 . . . . . . . . . 10 (𝐹 Fn {A} → (𝐹 “ dom 𝐹) = (𝐹 “ {A}))
1714, 16syl5eqr 2068 . . . . . . . . 9 (𝐹 Fn {A} → ran 𝐹 = (𝐹 “ {A}))
18 fnsnfv 5157 . . . . . . . . . 10 ((𝐹 Fn {A} A {A}) → {(𝐹A)} = (𝐹 “ {A}))
193, 18mpan2 403 . . . . . . . . 9 (𝐹 Fn {A} → {(𝐹A)} = (𝐹 “ {A}))
2017, 19eqtr4d 2057 . . . . . . . 8 (𝐹 Fn {A} → ran 𝐹 = {(𝐹A)})
21 feq3 4958 . . . . . . . 8 (ran 𝐹 = {(𝐹A)} → (𝐹:{A}⟶ran 𝐹𝐹:{A}⟶{(𝐹A)}))
2220, 21syl 14 . . . . . . 7 (𝐹 Fn {A} → (𝐹:{A}⟶ran 𝐹𝐹:{A}⟶{(𝐹A)}))
2313, 22mpbid 135 . . . . . 6 (𝐹 Fn {A} → 𝐹:{A}⟶{(𝐹A)})
241, 23syl 14 . . . . 5 (𝐹:{A}⟶B𝐹:{A}⟶{(𝐹A)})
2511, 24jca 290 . . . 4 (𝐹:{A}⟶B → ((𝐹A) B 𝐹:{A}⟶{(𝐹A)}))
26 snssi 3482 . . . . 5 ((𝐹A) B → {(𝐹A)} ⊆ B)
27 fss 4980 . . . . . 6 ((𝐹:{A}⟶{(𝐹A)} {(𝐹A)} ⊆ B) → 𝐹:{A}⟶B)
2827ancoms 255 . . . . 5 (({(𝐹A)} ⊆ B 𝐹:{A}⟶{(𝐹A)}) → 𝐹:{A}⟶B)
2926, 28sylan 267 . . . 4 (((𝐹A) B 𝐹:{A}⟶{(𝐹A)}) → 𝐹:{A}⟶B)
3025, 29impbii 117 . . 3 (𝐹:{A}⟶B ↔ ((𝐹A) B 𝐹:{A}⟶{(𝐹A)}))
31 fsng 5261 . . . . 5 ((A V (𝐹A) V) → (𝐹:{A}⟶{(𝐹A)} ↔ 𝐹 = {⟨A, (𝐹A)⟩}))
322, 31mpan 402 . . . 4 ((𝐹A) V → (𝐹:{A}⟶{(𝐹A)} ↔ 𝐹 = {⟨A, (𝐹A)⟩}))
3332anbi2d 440 . . 3 ((𝐹A) V → (((𝐹A) B 𝐹:{A}⟶{(𝐹A)}) ↔ ((𝐹A) B 𝐹 = {⟨A, (𝐹A)⟩})))
3430, 33syl5bb 181 . 2 ((𝐹A) V → (𝐹:{A}⟶B ↔ ((𝐹A) B 𝐹 = {⟨A, (𝐹A)⟩})))
357, 9, 34pm5.21nii 607 1 (𝐹:{A}⟶B ↔ ((𝐹A) B 𝐹 = {⟨A, (𝐹A)⟩}))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1228   wcel 1374  Vcvv 2535  wss 2894  {csn 3350  cop 3353  dom cdm 4272  ran crn 4273  cima 4275   Fn wfn 4824  wf 4825  cfv 4829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-reu 2291  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837
This theorem is referenced by:  fnressn  5274  fressnfv  5275
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