Theorem funcnvsn 6408

Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 6411 via cnvsn 6069, 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 5952	. 2 ⊢ Rel ◡{⟨𝐴, 𝐵⟩}
2		moeq 3609	. . . 4 ⊢ ∃*𝑦 𝑦 = 𝐴
3		vex 3402	. . . . . . . 8 ⊢ 𝑥 ∈ V
4		vex 3402	. . . . . . . 8 ⊢ 𝑦 ∈ V
5	3, 4	brcnv 5736	. . . . . . 7 ⊢ (𝑥◡{⟨𝐴, 𝐵⟩}𝑦 ↔ 𝑦{⟨𝐴, 𝐵⟩}𝑥)
6		df-br 5040	. . . . . . 7 ⊢ (𝑦{⟨𝐴, 𝐵⟩}𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
7	5, 6	bitri 278	. . . . . 6 ⊢ (𝑥◡{⟨𝐴, 𝐵⟩}𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
8		elsni 4544	. . . . . . 7 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩)
9	4, 3	opth1 5344	. . . . . . 7 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ → 𝑦 = 𝐴)
10	8, 9	syl 17	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → 𝑦 = 𝐴)
11	7, 10	sylbi 220	. . . . 5 ⊢ (𝑥◡{⟨𝐴, 𝐵⟩}𝑦 → 𝑦 = 𝐴)
12	11	moimi 2544	. . . 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 1803	. 2 ⊢ ∀𝑥∃*𝑦 𝑥◡{⟨𝐴, 𝐵⟩}𝑦
15		dffun6 6373	. 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 1541   = wceq 1543   ∈ wcel 2112  ∃*wmo 2537  {csn 4527  ⟨cop 4533   class class class wbr 5039  ^◡ccnv 5535  Rel wrel 5541  Fun wfun 6352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ral 3056  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-br 5040  df-opab 5102  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-fun 6360

This theorem is referenced by:  funsng  6409  funcnvpr  6420  funcnvtp  6421  funcnvs1  14442  0spth  28163  funen1cnv  32727