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Theorem funop 7158
Description: An ordered pair is a function iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) A function is a class of ordered pairs, so the fact that an ordered pair may sometimes be itself a function is an "accident" depending on the specific encoding of ordered pairs as classes (in set.mm, the Kuratowski encoding). A more meaningful statement is funsng 6604, as relsnopg 5805 is to relop 5853. (New usage is discouraged.)
Hypotheses
Ref Expression
funopsn.x 𝑋 ∈ V
funopsn.y 𝑌 ∈ V
Assertion
Ref Expression
funop (Fun ⟨𝑋, 𝑌⟩ ↔ ∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}))
Distinct variable groups:   𝑋,𝑎   𝑌,𝑎

Proof of Theorem funop
StepHypRef Expression
1 eqid 2728 . . 3 𝑋, 𝑌⟩ = ⟨𝑋, 𝑌
2 funopsn.x . . . 4 𝑋 ∈ V
3 funopsn.y . . . 4 𝑌 ∈ V
42, 3funopsn 7157 . . 3 ((Fun ⟨𝑋, 𝑌⟩ ∧ ⟨𝑋, 𝑌⟩ = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}))
51, 4mpan2 690 . 2 (Fun ⟨𝑋, 𝑌⟩ → ∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}))
6 vex 3475 . . . . . 6 𝑎 ∈ V
76, 6funsn 6606 . . . . 5 Fun {⟨𝑎, 𝑎⟩}
8 funeq 6573 . . . . 5 (⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩} → (Fun ⟨𝑋, 𝑌⟩ ↔ Fun {⟨𝑎, 𝑎⟩}))
97, 8mpbiri 258 . . . 4 (⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩} → Fun ⟨𝑋, 𝑌⟩)
109adantl 481 . . 3 ((𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}) → Fun ⟨𝑋, 𝑌⟩)
1110exlimiv 1926 . 2 (∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}) → Fun ⟨𝑋, 𝑌⟩)
125, 11impbii 208 1 (Fun ⟨𝑋, 𝑌⟩ ↔ ∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1534  wex 1774  wcel 2099  Vcvv 3471  {csn 4629  cop 4635  Fun wfun 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556
This theorem is referenced by:  funopdmsn  7159  funsndifnop  7160
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