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Theorem funop1 40327
Description: A function is an ordered pair iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.)
Assertion
Ref Expression
funop1 (∃𝑥𝑦 𝐹 = ⟨𝑥, 𝑦⟩ → (Fun 𝐹 ↔ ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
Distinct variable group:   𝑥,𝐹,𝑦

Proof of Theorem funop1
Dummy variables 𝑎 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq12 4342 . . . 4 ((𝑥 = 𝑣𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑣, 𝑤⟩)
21eqeq2d 2620 . . 3 ((𝑥 = 𝑣𝑦 = 𝑤) → (𝐹 = ⟨𝑥, 𝑦⟩ ↔ 𝐹 = ⟨𝑣, 𝑤⟩))
32cbvex2v 2275 . 2 (∃𝑥𝑦 𝐹 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑣𝑤 𝐹 = ⟨𝑣, 𝑤⟩)
4 vex 3176 . . . . . . 7 𝑣 ∈ V
5 vex 3176 . . . . . . 7 𝑤 ∈ V
64, 5funopsn 6319 . . . . . 6 ((Fun 𝐹𝐹 = ⟨𝑣, 𝑤⟩) → ∃𝑎(𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
7 vex 3176 . . . . . . . . 9 𝑎 ∈ V
8 opeq12 4342 . . . . . . . . . . 11 ((𝑥 = 𝑎𝑦 = 𝑎) → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑎⟩)
98sneqd 4137 . . . . . . . . . 10 ((𝑥 = 𝑎𝑦 = 𝑎) → {⟨𝑥, 𝑦⟩} = {⟨𝑎, 𝑎⟩})
109eqeq2d 2620 . . . . . . . . 9 ((𝑥 = 𝑎𝑦 = 𝑎) → (𝐹 = {⟨𝑥, 𝑦⟩} ↔ 𝐹 = {⟨𝑎, 𝑎⟩}))
117, 7, 10spc2ev 3274 . . . . . . . 8 (𝐹 = {⟨𝑎, 𝑎⟩} → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
1211adantl 481 . . . . . . 7 ((𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}) → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
1312exlimiv 1845 . . . . . 6 (∃𝑎(𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}) → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
146, 13syl 17 . . . . 5 ((Fun 𝐹𝐹 = ⟨𝑣, 𝑤⟩) → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
1514expcom 450 . . . 4 (𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
16 vex 3176 . . . . . . 7 𝑥 ∈ V
17 vex 3176 . . . . . . 7 𝑦 ∈ V
1816, 17funsn 5853 . . . . . 6 Fun {⟨𝑥, 𝑦⟩}
19 funeq 5823 . . . . . 6 (𝐹 = {⟨𝑥, 𝑦⟩} → (Fun 𝐹 ↔ Fun {⟨𝑥, 𝑦⟩}))
2018, 19mpbiri 247 . . . . 5 (𝐹 = {⟨𝑥, 𝑦⟩} → Fun 𝐹)
2120exlimivv 1847 . . . 4 (∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩} → Fun 𝐹)
2215, 21impbid1 214 . . 3 (𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 ↔ ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
2322exlimivv 1847 . 2 (∃𝑣𝑤 𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 ↔ ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
243, 23sylbi 206 1 (∃𝑥𝑦 𝐹 = ⟨𝑥, 𝑦⟩ → (Fun 𝐹 ↔ ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wex 1695  {csn 4125  cop 4131  Fun wfun 5798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812
This theorem is referenced by: (None)
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