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Theorem funcnv 5393

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 5392 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 2804	. . . . . . 7 ⊢ 𝑥 ∈ V
2		vex 2804	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	brelrn 4967	. . . . . 6 ⊢ (𝑥𝐴𝑦 → 𝑦 ∈ ran 𝐴)
4	3	pm4.71ri 392	. . . . 5 ⊢ (𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
5	4	mobii 2115	. . . 4 ⊢ (∃𝑥 𝑥𝐴𝑦 ↔ ∃𝑥(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
6		moanimv 2154	. . . 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 1518	. 2 ⊢ (∀𝑦∃𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦))
9		funcnv2 5392	. 2 ⊢ (Fun ^◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦)
10		df-ral 2514	. 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 1395 ∃*wmo 2079 ∈ wcel 2201 ∀wral 2509 class class class wbr 4089 ^◡ccnv 4726 ran crn 4728 Fun wfun 5322

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-14 2204 ax-ext 2212 ax-sep 4208 ax-pow 4266 ax-pr 4301

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-nf 1509 df-sb 1810 df-eu 2081 df-mo 2082 df-clab 2217 df-cleq 2223 df-clel 2226 df-nfc 2362 df-ral 2514 df-rex 2515 df-v 2803 df-un 3203 df-in 3205 df-ss 3212 df-pw 3655 df-sn 3676 df-pr 3677 df-op 3679 df-br 4090 df-opab 4152 df-id 4392 df-xp 4733 df-rel 4734 df-cnv 4735 df-co 4736 df-dm 4737 df-rn 4738 df-fun 5330

This theorem is referenced by: funcnv3 5394 fncnv 5398

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