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Theorem funop 7096
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 6553, as relsnopg 5760 is to relop 5807. (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 2733 . . 3 𝑋, 𝑌⟩ = ⟨𝑋, 𝑌
2 funopsn.x . . . 4 𝑋 ∈ V
3 funopsn.y . . . 4 𝑌 ∈ V
42, 3funopsn 7095 . . 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 3448 . . . . . 6 𝑎 ∈ V
76, 6funsn 6555 . . . . 5 Fun {⟨𝑎, 𝑎⟩}
8 funeq 6522 . . . . 5 (⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩} → (Fun ⟨𝑋, 𝑌⟩ ↔ Fun {⟨𝑎, 𝑎⟩}))
97, 8mpbiri 258 . . . 4 (⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩} → Fun ⟨𝑋, 𝑌⟩)
109adantl 483 . . 3 ((𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}) → Fun ⟨𝑋, 𝑌⟩)
1110exlimiv 1934 . 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 397   = wceq 1542  wex 1782  wcel 2107  Vcvv 3444  {csn 4587  cop 4593  Fun wfun 6491
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 5257  ax-nul 5264  ax-pr 5385
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505
This theorem is referenced by:  funopdmsn  7097  funsndifnop  7098
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