Theorem funcnvsn 5280

Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 5283 via cnvsn 5129, but stating it this way allows us to skip the sethood assumptions on 𝐴 and 𝐵. (Contributed by NM, 30-Apr-2015.)

Assertion

Ref	Expression
funcnvsn	⊢ Fun ◡{⟨𝐴, 𝐵⟩}

Proof of Theorem funcnvsn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcnv 5024	. 2 ⊢ Rel ◡{⟨𝐴, 𝐵⟩}
2		moeq 2927	. . . 4 ⊢ ∃*𝑦 𝑦 = 𝐴
3		vex 2755	. . . . . . . 8 ⊢ 𝑥 ∈ V
4		vex 2755	. . . . . . . 8 ⊢ 𝑦 ∈ V
5	3, 4	brcnv 4828	. . . . . . 7 ⊢ (𝑥◡{⟨𝐴, 𝐵⟩}𝑦 ↔ 𝑦{⟨𝐴, 𝐵⟩}𝑥)
6		df-br 4019	. . . . . . 7 ⊢ (𝑦{⟨𝐴, 𝐵⟩}𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
7	5, 6	bitri 184	. . . . . 6 ⊢ (𝑥◡{⟨𝐴, 𝐵⟩}𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
8		elsni 3625	. . . . . . 7 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩)
9	4, 3	opth1 4254	. . . . . . 7 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ → 𝑦 = 𝐴)
10	8, 9	syl 14	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → 𝑦 = 𝐴)
11	7, 10	sylbi 121	. . . . 5 ⊢ (𝑥◡{⟨𝐴, 𝐵⟩}𝑦 → 𝑦 = 𝐴)
12	11	moimi 2103	. . . 4 ⊢ (∃*𝑦 𝑦 = 𝐴 → ∃*𝑦 𝑥◡{⟨𝐴, 𝐵⟩}𝑦)
13	2, 12	ax-mp 5	. . 3 ⊢ ∃*𝑦 𝑥◡{⟨𝐴, 𝐵⟩}𝑦
14	13	ax-gen 1460	. 2 ⊢ ∀𝑥∃*𝑦 𝑥◡{⟨𝐴, 𝐵⟩}𝑦
15		dffun6 5249	. 2 ⊢ (Fun ◡{⟨𝐴, 𝐵⟩} ↔ (Rel ◡{⟨𝐴, 𝐵⟩} ∧ ∀𝑥∃*𝑦 𝑥◡{⟨𝐴, 𝐵⟩}𝑦))
16	1, 14, 15	mpbir2an 944	1 ⊢ Fun ◡{⟨𝐴, 𝐵⟩}

Syntax hints:  ∀wal 1362   = wceq 1364  ∃*wmo 2039   ∈ wcel 2160  {csn 3607  ⟨cop 3610   class class class wbr 4018  ^◡ccnv 4643  Rel wrel 4649  Fun wfun 5229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-fun 5237

This theorem is referenced by:  funsng  5281