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Theorem funop1 46039
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 4876 . . . 4 ((𝑥 = 𝑣𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑣, 𝑤⟩)
21eqeq2d 2744 . . 3 ((𝑥 = 𝑣𝑦 = 𝑤) → (𝐹 = ⟨𝑥, 𝑦⟩ ↔ 𝐹 = ⟨𝑣, 𝑤⟩))
32cbvex2vw 2045 . 2 (∃𝑥𝑦 𝐹 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑣𝑤 𝐹 = ⟨𝑣, 𝑤⟩)
4 vex 3479 . . . . . . 7 𝑣 ∈ V
5 vex 3479 . . . . . . 7 𝑤 ∈ V
64, 5funopsn 7146 . . . . . 6 ((Fun 𝐹𝐹 = ⟨𝑣, 𝑤⟩) → ∃𝑎(𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
7 vex 3479 . . . . . . . . 9 𝑎 ∈ V
8 opeq12 4876 . . . . . . . . . . 11 ((𝑥 = 𝑎𝑦 = 𝑎) → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑎⟩)
98sneqd 4641 . . . . . . . . . 10 ((𝑥 = 𝑎𝑦 = 𝑎) → {⟨𝑥, 𝑦⟩} = {⟨𝑎, 𝑎⟩})
109eqeq2d 2744 . . . . . . . . 9 ((𝑥 = 𝑎𝑦 = 𝑎) → (𝐹 = {⟨𝑥, 𝑦⟩} ↔ 𝐹 = {⟨𝑎, 𝑎⟩}))
117, 7, 10spc2ev 3598 . . . . . . . 8 (𝐹 = {⟨𝑎, 𝑎⟩} → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
1211adantl 483 . . . . . . 7 ((𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}) → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
1312exlimiv 1934 . . . . . 6 (∃𝑎(𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}) → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
146, 13syl 17 . . . . 5 ((Fun 𝐹𝐹 = ⟨𝑣, 𝑤⟩) → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
1514expcom 415 . . . 4 (𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
16 vex 3479 . . . . . . 7 𝑥 ∈ V
17 vex 3479 . . . . . . 7 𝑦 ∈ V
1816, 17funsn 6602 . . . . . 6 Fun {⟨𝑥, 𝑦⟩}
19 funeq 6569 . . . . . 6 (𝐹 = {⟨𝑥, 𝑦⟩} → (Fun 𝐹 ↔ Fun {⟨𝑥, 𝑦⟩}))
2018, 19mpbiri 258 . . . . 5 (𝐹 = {⟨𝑥, 𝑦⟩} → Fun 𝐹)
2120exlimivv 1936 . . . 4 (∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩} → Fun 𝐹)
2215, 21impbid1 224 . . 3 (𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 ↔ ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
2322exlimivv 1936 . 2 (∃𝑣𝑤 𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 ↔ ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
243, 23sylbi 216 1 (∃𝑥𝑦 𝐹 = ⟨𝑥, 𝑦⟩ → (Fun 𝐹 ↔ ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  {csn 4629  cop 4635  Fun wfun 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552
This theorem is referenced by: (None)
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