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Theorem funop1 43483
 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 4804 . . . 4 ((𝑥 = 𝑣𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑣, 𝑤⟩)
21eqeq2d 2832 . . 3 ((𝑥 = 𝑣𝑦 = 𝑤) → (𝐹 = ⟨𝑥, 𝑦⟩ ↔ 𝐹 = ⟨𝑣, 𝑤⟩))
32cbvex2vw 2044 . 2 (∃𝑥𝑦 𝐹 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑣𝑤 𝐹 = ⟨𝑣, 𝑤⟩)
4 vex 3497 . . . . . . 7 𝑣 ∈ V
5 vex 3497 . . . . . . 7 𝑤 ∈ V
64, 5funopsn 6909 . . . . . 6 ((Fun 𝐹𝐹 = ⟨𝑣, 𝑤⟩) → ∃𝑎(𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
7 vex 3497 . . . . . . . . 9 𝑎 ∈ V
8 opeq12 4804 . . . . . . . . . . 11 ((𝑥 = 𝑎𝑦 = 𝑎) → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑎⟩)
98sneqd 4578 . . . . . . . . . 10 ((𝑥 = 𝑎𝑦 = 𝑎) → {⟨𝑥, 𝑦⟩} = {⟨𝑎, 𝑎⟩})
109eqeq2d 2832 . . . . . . . . 9 ((𝑥 = 𝑎𝑦 = 𝑎) → (𝐹 = {⟨𝑥, 𝑦⟩} ↔ 𝐹 = {⟨𝑎, 𝑎⟩}))
117, 7, 10spc2ev 3607 . . . . . . . 8 (𝐹 = {⟨𝑎, 𝑎⟩} → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
1211adantl 484 . . . . . . 7 ((𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}) → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
1312exlimiv 1927 . . . . . 6 (∃𝑎(𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}) → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
146, 13syl 17 . . . . 5 ((Fun 𝐹𝐹 = ⟨𝑣, 𝑤⟩) → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
1514expcom 416 . . . 4 (𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
16 vex 3497 . . . . . . 7 𝑥 ∈ V
17 vex 3497 . . . . . . 7 𝑦 ∈ V
1816, 17funsn 6406 . . . . . 6 Fun {⟨𝑥, 𝑦⟩}
19 funeq 6374 . . . . . 6 (𝐹 = {⟨𝑥, 𝑦⟩} → (Fun 𝐹 ↔ Fun {⟨𝑥, 𝑦⟩}))
2018, 19mpbiri 260 . . . . 5 (𝐹 = {⟨𝑥, 𝑦⟩} → Fun 𝐹)
2120exlimivv 1929 . . . 4 (∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩} → Fun 𝐹)
2215, 21impbid1 227 . . 3 (𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 ↔ ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
2322exlimivv 1929 . 2 (∃𝑣𝑤 𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 ↔ ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
243, 23sylbi 219 1 (∃𝑥𝑦 𝐹 = ⟨𝑥, 𝑦⟩ → (Fun 𝐹 ↔ ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776  {csn 4566  ⟨cop 4572  Fun wfun 6348 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362 This theorem is referenced by: (None)
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