Theorem funop 7127

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 6567, as relsnopg 5772 is to relop 5818. (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 2761	. . 3 ⊢ ⟨𝑋, 𝑌⟩ = ⟨𝑋, 𝑌⟩
2		funopsn.x	. . . 4 ⊢ 𝑋 ∈ V
3		funopsn.y	. . . 4 ⊢ 𝑌 ∈ V
4	2, 3	funopsn 7125	. . 3 ⊢ ((Fun ⟨𝑋, 𝑌⟩ ∧ ⟨𝑋, 𝑌⟩ = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}))
5	1, 4	mpan2 701	. 2 ⊢ (Fun ⟨𝑋, 𝑌⟩ → ∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}))
6		vex 3457	. . . . . 6 ⊢ 𝑎 ∈ V
7	6, 6	funsn 6569	. . . . 5 ⊢ Fun {⟨𝑎, 𝑎⟩}
8		funeq 6536	. . . . 5 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩} → (Fun ⟨𝑋, 𝑌⟩ ↔ Fun {⟨𝑎, 𝑎⟩}))
9	7, 8	mpbiri 260	. . . 4 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩} → Fun ⟨𝑋, 𝑌⟩)
10	9	adantl 485	. . 3 ⊢ ((𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}) → Fun ⟨𝑋, 𝑌⟩)
11	10	exlimiv 1949	. 2 ⊢ (∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}) → Fun ⟨𝑋, 𝑌⟩)
12	5, 11	impbii 211	1 ⊢ (Fun ⟨𝑋, 𝑌⟩ ↔ ∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  Vcvv 3453  {csn 4579  ⟨cop 4585  Fun wfun 6510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524

This theorem is referenced by:  funopdmsn  7128  funsndifnop  7129