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Theorem fsng 5279
 Description: A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.)
Assertion
Ref Expression
fsng ((A 𝐶 B 𝐷) → (𝐹:{A}⟶{B} ↔ 𝐹 = {⟨A, B⟩}))

Proof of Theorem fsng
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sneq 3378 . . . 4 (𝑎 = A → {𝑎} = {A})
21feq2d 4978 . . 3 (𝑎 = A → (𝐹:{𝑎}⟶{𝑏} ↔ 𝐹:{A}⟶{𝑏}))
3 opeq1 3540 . . . . 5 (𝑎 = A → ⟨𝑎, 𝑏⟩ = ⟨A, 𝑏⟩)
43sneqd 3380 . . . 4 (𝑎 = A → {⟨𝑎, 𝑏⟩} = {⟨A, 𝑏⟩})
54eqeq2d 2048 . . 3 (𝑎 = A → (𝐹 = {⟨𝑎, 𝑏⟩} ↔ 𝐹 = {⟨A, 𝑏⟩}))
62, 5bibi12d 224 . 2 (𝑎 = A → ((𝐹:{𝑎}⟶{𝑏} ↔ 𝐹 = {⟨𝑎, 𝑏⟩}) ↔ (𝐹:{A}⟶{𝑏} ↔ 𝐹 = {⟨A, 𝑏⟩})))
7 sneq 3378 . . . 4 (𝑏 = B → {𝑏} = {B})
8 feq3 4975 . . . 4 ({𝑏} = {B} → (𝐹:{A}⟶{𝑏} ↔ 𝐹:{A}⟶{B}))
97, 8syl 14 . . 3 (𝑏 = B → (𝐹:{A}⟶{𝑏} ↔ 𝐹:{A}⟶{B}))
10 opeq2 3541 . . . . 5 (𝑏 = B → ⟨A, 𝑏⟩ = ⟨A, B⟩)
1110sneqd 3380 . . . 4 (𝑏 = B → {⟨A, 𝑏⟩} = {⟨A, B⟩})
1211eqeq2d 2048 . . 3 (𝑏 = B → (𝐹 = {⟨A, 𝑏⟩} ↔ 𝐹 = {⟨A, B⟩}))
139, 12bibi12d 224 . 2 (𝑏 = B → ((𝐹:{A}⟶{𝑏} ↔ 𝐹 = {⟨A, 𝑏⟩}) ↔ (𝐹:{A}⟶{B} ↔ 𝐹 = {⟨A, B⟩})))
14 vex 2554 . . 3 𝑎 V
15 vex 2554 . . 3 𝑏 V
1614, 15fsn 5278 . 2 (𝐹:{𝑎}⟶{𝑏} ↔ 𝐹 = {⟨𝑎, 𝑏⟩})
176, 13, 16vtocl2g 2611 1 ((A 𝐶 B 𝐷) → (𝐹:{A}⟶{B} ↔ 𝐹 = {⟨A, B⟩}))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {csn 3367  ⟨cop 3370  ⟶wf 4841 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-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  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-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852 This theorem is referenced by:  fsn2  5280  xpsng  5281  ftpg  5290  fseq1p1m1  8726
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