Theorem funop 7185

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 6621, as relsnopg 5790 is to relop 5838. (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 2763	. . 3 ⊢ ⟨𝑋, 𝑌⟩ = ⟨𝑋, 𝑌⟩
2		funopsn.x	. . . 4 ⊢ 𝑋 ∈ V
3		funopsn.y	. . . 4 ⊢ 𝑌 ∈ V
4	2, 3	funopsn 7183	. . 3 ⊢ ((Fun ⟨𝑋, 𝑌⟩ ∧ ⟨𝑋, 𝑌⟩ = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}))
5	1, 4	mpan2 704	. 2 ⊢ (Fun ⟨𝑋, 𝑌⟩ → ∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}))
6		vex 3458	. . . . . 6 ⊢ 𝑎 ∈ V
7	6, 6	funsn 6623	. . . . 5 ⊢ Fun {⟨𝑎, 𝑎⟩}
8		funeq 6587	. . . . 5 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩} → (Fun ⟨𝑋, 𝑌⟩ ↔ Fun {⟨𝑎, 𝑎⟩}))
9	7, 8	mpbiri 261	. . . 4 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩} → Fun ⟨𝑋, 𝑌⟩)
10	9	adantl 487	. . 3 ⊢ ((𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}) → Fun ⟨𝑋, 𝑌⟩)
11	10	exlimiv 1965	. 2 ⊢ (∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}) → Fun ⟨𝑋, 𝑌⟩)
12	5, 11	impbii 212	1 ⊢ (Fun ⟨𝑋, 𝑌⟩ ↔ ∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  Vcvv 3454  {csn 4588  ⟨cop 4840  Fun wfun 6560

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574

This theorem is used by:  funopdmsn  7186  funsndifnop  7187