Theorem fun2cnv 5310

Description: The double converse of a class is a function iff the class is single-valued. Each side is equivalent to Definition 6.4(2) of [TakeutiZaring] p. 23, who use the notation "Un(A)" for single-valued. Note that 𝐴 is not necessarily a function. (Contributed by NM, 13-Aug-2004.)

Assertion

Ref	Expression
fun2cnv	⊢ (Fun ◡◡𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem fun2cnv

Step	Hyp	Ref	Expression
1		funcnv2 5306	. 2 ⊢ (Fun ◡◡𝐴 ↔ ∀𝑥∃*𝑦 𝑦◡𝐴𝑥)
2		vex 2763	. . . . 5 ⊢ 𝑦 ∈ V
3		vex 2763	. . . . 5 ⊢ 𝑥 ∈ V
4	2, 3	brcnv 4839	. . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
5	4	mobii 2079	. . 3 ⊢ (∃*𝑦 𝑦◡𝐴𝑥 ↔ ∃*𝑦 𝑥𝐴𝑦)
6	5	albii 1481	. 2 ⊢ (∀𝑥∃*𝑦 𝑦◡𝐴𝑥 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
7	1, 6	bitri 184	1 ⊢ (Fun ◡◡𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)

Syntax hints:   ↔ wb 105  ∀wal 1362  ∃*wmo 2043   class class class wbr 4029  ^◡ccnv 4654  Fun wfun 5240

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 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-fun 5248

This theorem is referenced by:  svrelfun  5311  fun11  5313