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Theorem funcnv 4985

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 4984 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 2575	. . . . . . 7 ⊢ 𝑥 ∈ V
2		vex 2575	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	brelrn 4592	. . . . . 6 ⊢ (𝑥𝐴𝑦 → 𝑦 ∈ ran 𝐴)
4	3	pm4.71ri 378	. . . . 5 ⊢ (𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
5	4	mobii 1951	. . . 4 ⊢ (∃𝑥 𝑥𝐴𝑦 ↔ ∃𝑥(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
6		moanimv 1989	. . . 4 ⊢ (∃𝑥(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦) ↔ (𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦))
7	5, 6	bitri 177	. . 3 ⊢ (∃𝑥 𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦))
8	7	albii 1373	. 2 ⊢ (∀𝑦∃𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦))
9		funcnv2 4984	. 2 ⊢ (Fun ^◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦)
10		df-ral 2326	. 2 ⊢ (∀𝑦 ∈ ran 𝐴∃𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦))
11	8, 9, 10	3bitr4i 205	1 ⊢ (Fun ^◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 101 ↔ wb 102 ∀wal 1255 ∈ wcel 1407 ∃*wmo 1915 ∀wral 2321 class class class wbr 3789 ^◡ccnv 4369 ran crn 4371 Fun wfun 4921

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 103 ax-ia2 104 ax-ia3 105 ax-io 638 ax-5 1350 ax-7 1351 ax-gen 1352 ax-ie1 1396 ax-ie2 1397 ax-8 1409 ax-10 1410 ax-11 1411 ax-i12 1412 ax-bndl 1413 ax-4 1414 ax-14 1419 ax-17 1433 ax-i9 1437 ax-ial 1441 ax-i5r 1442 ax-ext 2036 ax-sep 3900 ax-pow 3952 ax-pr 3969

This theorem depends on definitions: df-bi 114 df-3an 896 df-tru 1260 df-nf 1364 df-sb 1660 df-eu 1917 df-mo 1918 df-clab 2041 df-cleq 2047 df-clel 2050 df-nfc 2181 df-ral 2326 df-rex 2327 df-v 2574 df-un 2947 df-in 2949 df-ss 2956 df-pw 3386 df-sn 3406 df-pr 3407 df-op 3409 df-br 3790 df-opab 3844 df-id 4055 df-xp 4376 df-rel 4377 df-cnv 4378 df-co 4379 df-dm 4380 df-rn 4381 df-fun 4929

This theorem is referenced by: funcnv3 4986 fncnv 4990

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