Theorem fun2cnv 2704

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.

Assertion

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

Distinct variable group(s):   x,y,A

Proof of Theorem fun2cnv

Step	Hyp	Ref	Expression
1		funcnv2 2702	. 2 ⊢ (Fun ◡◡A ↔ ∀x∃*y y◡Ax)
2		visset 1350	. . . . 5 ⊢ y ∈ V
3		visset 1350	. . . . 5 ⊢ x ∈ V
4	2, 3	brcnv 2519	. . . 4 ⊢ (y◡Ax ↔ xAy)
5	4	bimo 1031	. . 3 ⊢ (∃*y y◡Ax ↔ ∃*y xAy)
6	5	bial 695	. 2 ⊢ (∀x∃*y y◡Ax ↔ ∀x∃*y xAy)
7	1, 6	bitr 151	1 ⊢ (Fun ◡◡A ↔ ∀x∃*y xAy)

Syntax hints:   ↔ wb 127  ∀wal 672  ∃*wmo 1008   class class class wbr 2054  ^◡ccnv 2409  Fun wfun 2416

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-fun 2432

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