Theorem funcnv 4980

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 4979 for a simpler version. (Contributed by NM, 13-Aug-2004.)

Assertion

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

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem funcnv

Step	Hyp	Ref	Expression
1		vex 2604	. . . . . . 7 ⊢ 𝑥 ∈ V
2		vex 2604	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	brelrn 4585	. . . . . 6 ⊢ (𝑥𝐴𝑦 → 𝑦 ∈ ran 𝐴)
4	3	pm4.71ri 384	. . . . 5 ⊢ (𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
5	4	mobii 1978	. . . 4 ⊢ (∃*𝑥 𝑥𝐴𝑦 ↔ ∃*𝑥(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
6		moanimv 2016	. . . 4 ⊢ (∃*𝑥(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦) ↔ (𝑦 ∈ ran 𝐴 → ∃*𝑥 𝑥𝐴𝑦))
7	5, 6	bitri 182	. . 3 ⊢ (∃*𝑥 𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 → ∃*𝑥 𝑥𝐴𝑦))
8	7	albii 1399	. 2 ⊢ (∀𝑦∃*𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(𝑦 ∈ ran 𝐴 → ∃*𝑥 𝑥𝐴𝑦))
9		funcnv2 4979	. 2 ⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦)
10		df-ral 2353	. 2 ⊢ (∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(𝑦 ∈ ran 𝐴 → ∃*𝑥 𝑥𝐴𝑦))
11	8, 9, 10	3bitr4i 210	1 ⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1282   ∈ wcel 1433  ∃*wmo 1942  ∀wral 2348   class class class wbr 3785  ^◡ccnv 4362  ran crn 4364  Fun wfun 4916

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-fun 4924

This theorem is referenced by:  funcnv3  4981  fncnv  4985