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Theorem funop1 47295
Description: A function is an ordered pair iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) (Avoid depending on this detail.)
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 4875 . . . 4 ((𝑥 = 𝑣𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑣, 𝑤⟩)
21eqeq2d 2748 . . 3 ((𝑥 = 𝑣𝑦 = 𝑤) → (𝐹 = ⟨𝑥, 𝑦⟩ ↔ 𝐹 = ⟨𝑣, 𝑤⟩))
32cbvex2vw 2040 . 2 (∃𝑥𝑦 𝐹 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑣𝑤 𝐹 = ⟨𝑣, 𝑤⟩)
4 vex 3484 . . . . . . 7 𝑣 ∈ V
5 vex 3484 . . . . . . 7 𝑤 ∈ V
64, 5funopsn 7168 . . . . . 6 ((Fun 𝐹𝐹 = ⟨𝑣, 𝑤⟩) → ∃𝑎(𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
7 vex 3484 . . . . . . . . 9 𝑎 ∈ V
8 opeq12 4875 . . . . . . . . . . 11 ((𝑥 = 𝑎𝑦 = 𝑎) → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑎⟩)
98sneqd 4638 . . . . . . . . . 10 ((𝑥 = 𝑎𝑦 = 𝑎) → {⟨𝑥, 𝑦⟩} = {⟨𝑎, 𝑎⟩})
109eqeq2d 2748 . . . . . . . . 9 ((𝑥 = 𝑎𝑦 = 𝑎) → (𝐹 = {⟨𝑥, 𝑦⟩} ↔ 𝐹 = {⟨𝑎, 𝑎⟩}))
117, 7, 10spc2ev 3607 . . . . . . . 8 (𝐹 = {⟨𝑎, 𝑎⟩} → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
1211adantl 481 . . . . . . 7 ((𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}) → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
1312exlimiv 1930 . . . . . 6 (∃𝑎(𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}) → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
146, 13syl 17 . . . . 5 ((Fun 𝐹𝐹 = ⟨𝑣, 𝑤⟩) → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
1514expcom 413 . . . 4 (𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
16 vex 3484 . . . . . . 7 𝑥 ∈ V
17 vex 3484 . . . . . . 7 𝑦 ∈ V
1816, 17funsn 6619 . . . . . 6 Fun {⟨𝑥, 𝑦⟩}
19 funeq 6586 . . . . . 6 (𝐹 = {⟨𝑥, 𝑦⟩} → (Fun 𝐹 ↔ Fun {⟨𝑥, 𝑦⟩}))
2018, 19mpbiri 258 . . . . 5 (𝐹 = {⟨𝑥, 𝑦⟩} → Fun 𝐹)
2120exlimivv 1932 . . . 4 (∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩} → Fun 𝐹)
2215, 21impbid1 225 . . 3 (𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 ↔ ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
2322exlimivv 1932 . 2 (∃𝑣𝑤 𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 ↔ ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
243, 23sylbi 217 1 (∃𝑥𝑦 𝐹 = ⟨𝑥, 𝑦⟩ → (Fun 𝐹 ↔ ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  {csn 4626  cop 4632  Fun wfun 6555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by: (None)
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