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Theorem fsng 5336
 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 ((𝐴𝐶𝐵𝐷) → (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 3386 . . . 4 (𝑎 = 𝐴 → {𝑎} = {𝐴})
21feq2d 5035 . . 3 (𝑎 = 𝐴 → (𝐹:{𝑎}⟶{𝑏} ↔ 𝐹:{𝐴}⟶{𝑏}))
3 opeq1 3549 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
43sneqd 3388 . . . 4 (𝑎 = 𝐴 → {⟨𝑎, 𝑏⟩} = {⟨𝐴, 𝑏⟩})
54eqeq2d 2051 . . 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 3386 . . . 4 (𝑏 = 𝐵 → {𝑏} = {𝐵})
8 feq3 5032 . . . 4 ({𝑏} = {𝐵} → (𝐹:{𝐴}⟶{𝑏} ↔ 𝐹:{𝐴}⟶{𝐵}))
97, 8syl 14 . . 3 (𝑏 = 𝐵 → (𝐹:{𝐴}⟶{𝑏} ↔ 𝐹:{𝐴}⟶{𝐵}))
10 opeq2 3550 . . . . 5 (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
1110sneqd 3388 . . . 4 (𝑏 = 𝐵 → {⟨𝐴, 𝑏⟩} = {⟨𝐴, 𝐵⟩})
1211eqeq2d 2051 . . 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 2560 . . 3 𝑎 ∈ V
15 vex 2560 . . 3 𝑏 ∈ V
1614, 15fsn 5335 . 2 (𝐹:{𝑎}⟶{𝑏} ↔ 𝐹 = {⟨𝑎, 𝑏⟩})
176, 13, 16vtocl2g 2617 1 ((𝐴𝐶𝐵𝐷) → (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩}))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  {csn 3375  ⟨cop 3378  ⟶wf 4898 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909 This theorem is referenced by:  fsn2  5337  xpsng  5338  ftpg  5347  fseq1p1m1  8956
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