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Theorem funcnv 5120

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 5119 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 2644	. . . . . . 7 ⊢ 𝑥 ∈ V
2		vex 2644	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	brelrn 4710	. . . . . 6 ⊢ (𝑥𝐴𝑦 → 𝑦 ∈ ran 𝐴)
4	3	pm4.71ri 387	. . . . 5 ⊢ (𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
5	4	mobii 1997	. . . 4 ⊢ (∃𝑥 𝑥𝐴𝑦 ↔ ∃𝑥(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
6		moanimv 2035	. . . 4 ⊢ (∃𝑥(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦) ↔ (𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦))
7	5, 6	bitri 183	. . 3 ⊢ (∃𝑥 𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦))
8	7	albii 1414	. 2 ⊢ (∀𝑦∃𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦))
9		funcnv2 5119	. 2 ⊢ (Fun ^◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦)
10		df-ral 2380	. 2 ⊢ (∀𝑦 ∈ ran 𝐴∃𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦))
11	8, 9, 10	3bitr4i 211	1 ⊢ (Fun ^◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1297 ∈ wcel 1448 ∃*wmo 1961 ∀wral 2375 class class class wbr 3875 ^◡ccnv 4476 ran crn 4478 Fun wfun 5053

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069

This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-br 3876 df-opab 3930 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-fun 5061

This theorem is referenced by: funcnv3 5121 fncnv 5125

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