Theorem fun2cnv 6603

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 6600	. 2 ⊢ (Fun ◡◡𝐴 ↔ ∀𝑥∃*𝑦 𝑦◡𝐴𝑥)
2		vex 3455	. . . . 5 ⊢ 𝑦 ∈ V
3		vex 3455	. . . . 5 ⊢ 𝑥 ∈ V
4	2, 3	brcnv 5860	. . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
5	4	mobii 2574	. . 3 ⊢ (∃*𝑦 𝑦◡𝐴𝑥 ↔ ∃*𝑦 𝑥𝐴𝑦)
6	5	albii 1852	. 2 ⊢ (∀𝑥∃*𝑦 𝑦◡𝐴𝑥 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
7	1, 6	bitri 278	1 ⊢ (Fun ◡◡𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1568  ∃*wmo 2563   class class class wbr 5103  ^◡ccnv 5650  Fun wfun 6525

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-fun 6533

This theorem is used by:  svrelfun  6604  fun11  6606