Theorem funcnv 2703

Description: The converse of a class is a function iff the class is single-rooted, which means that for any y in the range of A there is at most one x such that xAy. Definition of single-rooted in [Enderton] p. 43. See funcnv2 2702 for a simpler version.

Assertion

Ref	Expression
funcnv	⊢ (Fun ◡A ↔ ∀y ∈ ran A∃*x xAy)

Distinct variable group(s):   x,y,A

Proof of Theorem funcnv

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . . 7 ⊢ x ∈ V
2		visset 1350	. . . . . . 7 ⊢ y ∈ V
3	1, 2	brelrn 2559	. . . . . 6 ⊢ (xAy → y ∈ ran A)
4	3	pm4.71ri 484	. . . . 5 ⊢ (xAy ↔ (y ∈ ran A ∧ xAy))
5	4	bimo 1031	. . . 4 ⊢ (∃*x xAy ↔ ∃*x(y ∈ ran A ∧ xAy))
6		moanimv 1052	. . . 4 ⊢ (∃*x(y ∈ ran A ∧ xAy) ↔ (y ∈ ran A → ∃*x xAy))
7	5, 6	bitr 151	. . 3 ⊢ (∃*x xAy ↔ (y ∈ ran A → ∃*x xAy))
8	7	bial 695	. 2 ⊢ (∀y∃*x xAy ↔ ∀y(y ∈ ran A → ∃*x xAy))
9		funcnv2 2702	. 2 ⊢ (Fun ◡A ↔ ∀y∃*x xAy)
10		df-ral 1205	. 2 ⊢ (∀y ∈ ran A∃*x xAy ↔ ∀y(y ∈ ran A → ∃*x xAy))
11	8, 9, 10	3bitr4 158	1 ⊢ (Fun ◡A ↔ ∀y ∈ ran A∃*x xAy)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃*wmo 1008   ∈ wcel 1092  ∀wral 1201   class class class wbr 2054  ^◡ccnv 2409  ran crn 2411  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-ral 1205  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-dm 2428  df-rn 2429  df-fun 2432

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