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Theorem funcnv 5319

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 5318 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 2766	. . . . . . 7 ⊢ 𝑥 ∈ V
2		vex 2766	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	brelrn 4899	. . . . . 6 ⊢ (𝑥𝐴𝑦 → 𝑦 ∈ ran 𝐴)
4	3	pm4.71ri 392	. . . . 5 ⊢ (𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
5	4	mobii 2082	. . . 4 ⊢ (∃𝑥 𝑥𝐴𝑦 ↔ ∃𝑥(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
6		moanimv 2120	. . . 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 1484	. 2 ⊢ (∀𝑦∃𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦))
9		funcnv2 5318	. 2 ⊢ (Fun ^◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦)
10		df-ral 2480	. 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 2046 ∈ wcel 2167 ∀wral 2475 class class class wbr 4033 ^◡ccnv 4662 ran crn 4664 Fun wfun 5252

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ral 2480 df-rex 2481 df-v 2765 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-br 4034 df-opab 4095 df-id 4328 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-fun 5260

This theorem is referenced by: funcnv3 5320 fncnv 5324

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