Theorem funcnvsn 6560

Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 6563 via cnvsn 6202, 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 6083	. 2 ⊢ Rel ◡{⟨𝐴, 𝐵⟩}
2		moeq 3664	. . . 4 ⊢ ∃*𝑦 𝑦 = 𝐴
3		vex 3452	. . . . . . . 8 ⊢ 𝑥 ∈ V
4		vex 3452	. . . . . . . 8 ⊢ 𝑦 ∈ V
5	3, 4	brcnv 5847	. . . . . . 7 ⊢ (𝑥◡{⟨𝐴, 𝐵⟩}𝑦 ↔ 𝑦{⟨𝐴, 𝐵⟩}𝑥)
6		df-br 5095	. . . . . . 7 ⊢ (𝑦{⟨𝐴, 𝐵⟩}𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
7	5, 6	bitri 277	. . . . . 6 ⊢ (𝑥◡{⟨𝐴, 𝐵⟩}𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
8		elsni 4593	. . . . . . 7 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩)
9	4, 3	opth1 5437	. . . . . . 7 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ → 𝑦 = 𝐴)
10	8, 9	syl 17	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → 𝑦 = 𝐴)
11	7, 10	sylbi 219	. . . . 5 ⊢ (𝑥◡{⟨𝐴, 𝐵⟩}𝑦 → 𝑦 = 𝐴)
12	11	moimi 2566	. . . 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 1809	. 2 ⊢ ∀𝑥∃*𝑦 𝑥◡{⟨𝐴, 𝐵⟩}𝑦
15		dffun6 6521	. 2 ⊢ (Fun ◡{⟨𝐴, 𝐵⟩} ↔ (Rel ◡{⟨𝐴, 𝐵⟩} ∧ ∀𝑥∃*𝑦 𝑥◡{⟨𝐴, 𝐵⟩}𝑦))
16	1, 14, 15	mpbir2an 719	1 ⊢ Fun ◡{⟨𝐴, 𝐵⟩}

Syntax hints:  ∀wal 1552   = wceq 1554   ∈ wcel 2136  ∃*wmo 2558  {csn 4576  ⟨cop 4582   class class class wbr 5094  ^◡ccnv 5639  Rel wrel 5645  Fun wfun 6504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728  ax-sep 5240  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-mo 2560  df-clab 2735  df-cleq 2748  df-clel 2831  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-br 5095  df-opab 5157  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-fun 6512

This theorem is referenced by:  funsng  6561  funcnvpr  6572  funcnvtp  6573  funcnvs1  14915  0spth  30267  funen1cnv  35339