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Theorem funcnv 5315

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 5314 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 2763	. . . . . . 7 ⊢ 𝑥 ∈ V
2		vex 2763	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	brelrn 4895	. . . . . 6 ⊢ (𝑥𝐴𝑦 → 𝑦 ∈ ran 𝐴)
4	3	pm4.71ri 392	. . . . 5 ⊢ (𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
5	4	mobii 2079	. . . 4 ⊢ (∃𝑥 𝑥𝐴𝑦 ↔ ∃𝑥(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
6		moanimv 2117	. . . 4 ⊢ (∃𝑥(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦) ↔ (𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦))
7	5, 6	bitri 184	. . 3 ⊢ (∃𝑥 𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦))
8	7	albii 1481	. 2 ⊢ (∀𝑦∃𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦))
9		funcnv2 5314	. 2 ⊢ (Fun ^◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦)
10		df-ral 2477	. 2 ⊢ (∀𝑦 ∈ ran 𝐴∃𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦))
11	8, 9, 10	3bitr4i 212	1 ⊢ (Fun ^◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1362 ∃*wmo 2043 ∈ wcel 2164 ∀wral 2472 class class class wbr 4029 ^◡ccnv 4658 ran crn 4660 Fun wfun 5248

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-br 4030 df-opab 4091 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-fun 5256

This theorem is referenced by: funcnv3 5316 fncnv 5320

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