Theorem funcnvsn 6566

Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 6569 via cnvsn 6199, but stating it this way allows 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 6075	. 2 ⊢ Rel ◡{⟨𝐴, 𝐵⟩}
2		moeq 3678	. . . 4 ⊢ ∃*𝑦 𝑦 = 𝐴
3		vex 3451	. . . . . . . 8 ⊢ 𝑥 ∈ V
4		vex 3451	. . . . . . . 8 ⊢ 𝑦 ∈ V
5	3, 4	brcnv 5846	. . . . . . 7 ⊢ (𝑥◡{⟨𝐴, 𝐵⟩}𝑦 ↔ 𝑦{⟨𝐴, 𝐵⟩}𝑥)
6		df-br 5108	. . . . . . 7 ⊢ (𝑦{⟨𝐴, 𝐵⟩}𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
7	5, 6	bitri 275	. . . . . 6 ⊢ (𝑥◡{⟨𝐴, 𝐵⟩}𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
8		elsni 4606	. . . . . . 7 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩)
9	4, 3	opth1 5435	. . . . . . 7 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ → 𝑦 = 𝐴)
10	8, 9	syl 17	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → 𝑦 = 𝐴)
11	7, 10	sylbi 217	. . . . 5 ⊢ (𝑥◡{⟨𝐴, 𝐵⟩}𝑦 → 𝑦 = 𝐴)
12	11	moimi 2538	. . . 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 1795	. 2 ⊢ ∀𝑥∃*𝑦 𝑥◡{⟨𝐴, 𝐵⟩}𝑦
15		dffun6 6524	. 2 ⊢ (Fun ◡{⟨𝐴, 𝐵⟩} ↔ (Rel ◡{⟨𝐴, 𝐵⟩} ∧ ∀𝑥∃*𝑦 𝑥◡{⟨𝐴, 𝐵⟩}𝑦))
16	1, 14, 15	mpbir2an 711	1 ⊢ Fun ◡{⟨𝐴, 𝐵⟩}

Syntax hints:  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∃*wmo 2531  {csn 4589  ⟨cop 4595   class class class wbr 5107  ^◡ccnv 5637  Rel wrel 5643  Fun wfun 6505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-fun 6513

This theorem is referenced by:  funsng  6567  funcnvpr  6578  funcnvtp  6579  funcnvs1  14878  0spth  30055  funen1cnv  35078