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Theorem funop1 41818
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 4539 . . . 4 ((𝑥 = 𝑣𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑣, 𝑤⟩)
21eqeq2d 2780 . . 3 ((𝑥 = 𝑣𝑦 = 𝑤) → (𝐹 = ⟨𝑥, 𝑦⟩ ↔ 𝐹 = ⟨𝑣, 𝑤⟩))
32cbvex2v 2442 . 2 (∃𝑥𝑦 𝐹 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑣𝑤 𝐹 = ⟨𝑣, 𝑤⟩)
4 vex 3352 . . . . . . 7 𝑣 ∈ V
5 vex 3352 . . . . . . 7 𝑤 ∈ V
64, 5funopsn 6555 . . . . . 6 ((Fun 𝐹𝐹 = ⟨𝑣, 𝑤⟩) → ∃𝑎(𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
7 vex 3352 . . . . . . . . 9 𝑎 ∈ V
8 opeq12 4539 . . . . . . . . . . 11 ((𝑥 = 𝑎𝑦 = 𝑎) → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑎⟩)
98sneqd 4326 . . . . . . . . . 10 ((𝑥 = 𝑎𝑦 = 𝑎) → {⟨𝑥, 𝑦⟩} = {⟨𝑎, 𝑎⟩})
109eqeq2d 2780 . . . . . . . . 9 ((𝑥 = 𝑎𝑦 = 𝑎) → (𝐹 = {⟨𝑥, 𝑦⟩} ↔ 𝐹 = {⟨𝑎, 𝑎⟩}))
117, 7, 10spc2ev 3450 . . . . . . . 8 (𝐹 = {⟨𝑎, 𝑎⟩} → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
1211adantl 467 . . . . . . 7 ((𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}) → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
1312exlimiv 2009 . . . . . 6 (∃𝑎(𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}) → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
146, 13syl 17 . . . . 5 ((Fun 𝐹𝐹 = ⟨𝑣, 𝑤⟩) → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
1514expcom 398 . . . 4 (𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 → ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
16 vex 3352 . . . . . . 7 𝑥 ∈ V
17 vex 3352 . . . . . . 7 𝑦 ∈ V
1816, 17funsn 6082 . . . . . 6 Fun {⟨𝑥, 𝑦⟩}
19 funeq 6051 . . . . . 6 (𝐹 = {⟨𝑥, 𝑦⟩} → (Fun 𝐹 ↔ Fun {⟨𝑥, 𝑦⟩}))
2018, 19mpbiri 248 . . . . 5 (𝐹 = {⟨𝑥, 𝑦⟩} → Fun 𝐹)
2120exlimivv 2011 . . . 4 (∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩} → Fun 𝐹)
2215, 21impbid1 215 . . 3 (𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 ↔ ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
2322exlimivv 2011 . 2 (∃𝑣𝑤 𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 ↔ ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
243, 23sylbi 207 1 (∃𝑥𝑦 𝐹 = ⟨𝑥, 𝑦⟩ → (Fun 𝐹 ↔ ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wex 1851  {csn 4314  cop 4320  Fun wfun 6025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039
This theorem is referenced by: (None)
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