Theorem funop 7094

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 6543, as relsnopg 5752 is to relop 5799. (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

Step	Hyp	Ref	Expression
1		eqid 2736	. . 3 ⊢ ⟨𝑋, 𝑌⟩ = ⟨𝑋, 𝑌⟩
2		funopsn.x	. . . 4 ⊢ 𝑋 ∈ V
3		funopsn.y	. . . 4 ⊢ 𝑌 ∈ V
4	2, 3	funopsn 7093	. . 3 ⊢ ((Fun ⟨𝑋, 𝑌⟩ ∧ ⟨𝑋, 𝑌⟩ = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}))
5	1, 4	mpan2 691	. 2 ⊢ (Fun ⟨𝑋, 𝑌⟩ → ∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}))
6		vex 3444	. . . . . 6 ⊢ 𝑎 ∈ V
7	6, 6	funsn 6545	. . . . 5 ⊢ Fun {⟨𝑎, 𝑎⟩}
8		funeq 6512	. . . . 5 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩} → (Fun ⟨𝑋, 𝑌⟩ ↔ Fun {⟨𝑎, 𝑎⟩}))
9	7, 8	mpbiri 258	. . . 4 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩} → Fun ⟨𝑋, 𝑌⟩)
10	9	adantl 481	. . 3 ⊢ ((𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}) → Fun ⟨𝑋, 𝑌⟩)
11	10	exlimiv 1931	. 2 ⊢ (∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}) → Fun ⟨𝑋, 𝑌⟩)
12	5, 11	impbii 209	1 ⊢ (Fun ⟨𝑋, 𝑌⟩ ↔ ∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  Vcvv 3440  {csn 4580  ⟨cop 4586  Fun wfun 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500

This theorem is referenced by:  funopdmsn  7095  funsndifnop  7096