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Theorem fsn2 5280
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 4989 . . 3 (𝐹:{A}⟶B𝐹 Fn {A})
2 fsn2.1 . . . . 5 A V
32snid 3394 . . . 4 A {A}
4 funfvex 5135 . . . . 5 ((Fun 𝐹 A dom 𝐹) → (𝐹A) V)
54funfni 4942 . . . 4 ((𝐹 Fn {A} A {A}) → (𝐹A) V)
63, 5mpan2 401 . . 3 (𝐹 Fn {A} → (𝐹A) V)
71, 6syl 14 . 2 (𝐹:{A}⟶B → (𝐹A) V)
8 elex 2560 . . 3 ((𝐹A) B → (𝐹A) V)
98adantr 261 . 2 (((𝐹A) B 𝐹 = {⟨A, (𝐹A)⟩}) → (𝐹A) V)
10 ffvelrn 5243 . . . . . 6 ((𝐹:{A}⟶B A {A}) → (𝐹A) B)
113, 10mpan2 401 . . . . 5 (𝐹:{A}⟶B → (𝐹A) B)
12 dffn3 4996 . . . . . . . 8 (𝐹 Fn {A} ↔ 𝐹:{A}⟶ran 𝐹)
1312biimpi 113 . . . . . . 7 (𝐹 Fn {A} → 𝐹:{A}⟶ran 𝐹)
14 imadmrn 4621 . . . . . . . . . 10 (𝐹 “ dom 𝐹) = ran 𝐹
15 fndm 4941 . . . . . . . . . . 11 (𝐹 Fn {A} → dom 𝐹 = {A})
1615imaeq2d 4611 . . . . . . . . . 10 (𝐹 Fn {A} → (𝐹 “ dom 𝐹) = (𝐹 “ {A}))
1714, 16syl5eqr 2083 . . . . . . . . 9 (𝐹 Fn {A} → ran 𝐹 = (𝐹 “ {A}))
18 fnsnfv 5175 . . . . . . . . . 10 ((𝐹 Fn {A} A {A}) → {(𝐹A)} = (𝐹 “ {A}))
193, 18mpan2 401 . . . . . . . . 9 (𝐹 Fn {A} → {(𝐹A)} = (𝐹 “ {A}))
2017, 19eqtr4d 2072 . . . . . . . 8 (𝐹 Fn {A} → ran 𝐹 = {(𝐹A)})
21 feq3 4975 . . . . . . . 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 3499 . . . . 5 ((𝐹A) B → {(𝐹A)} ⊆ B)
27 fss 4997 . . . . . 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 5279 . . . . 5 ((A V (𝐹A) V) → (𝐹:{A}⟶{(𝐹A)} ↔ 𝐹 = {⟨A, (𝐹A)⟩}))
322, 31mpan 400 . . . 4 ((𝐹A) V → (𝐹:{A}⟶{(𝐹A)} ↔ 𝐹 = {⟨A, (𝐹A)⟩}))
3332anbi2d 437 . . 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 619 1 (𝐹:{A}⟶B ↔ ((𝐹A) B 𝐹 = {⟨A, (𝐹A)⟩}))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242   wcel 1390  Vcvv 2551  wss 2911  {csn 3367  cop 3370  dom cdm 4288  ran crn 4289  cima 4291   Fn wfn 4840  wf 4841  cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853
This theorem is referenced by:  fnressn  5292  fressnfv  5293  en1  6215
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