Theorem funcnv 5416

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 5415 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 2815	. . . . . . 7 ⊢ 𝑥 ∈ V
2		vex 2815	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	brelrn 4989	. . . . . 6 ⊢ (𝑥𝐴𝑦 → 𝑦 ∈ ran 𝐴)
4	3	pm4.71ri 392	. . . . 5 ⊢ (𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
5	4	mobii 2117	. . . 4 ⊢ (∃*𝑥 𝑥𝐴𝑦 ↔ ∃*𝑥(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
6		moanimv 2156	. . . 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 1519	. 2 ⊢ (∀𝑦∃*𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(𝑦 ∈ ran 𝐴 → ∃*𝑥 𝑥𝐴𝑦))
9		funcnv2 5415	. 2 ⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦)
10		df-ral 2525	. 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)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1396  ∃*wmo 2081   ∈ wcel 2203  ∀wral 2520   class class class wbr 4108  ^◡ccnv 4747  ran crn 4749  Fun wfun 5345

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-fun 5353

This theorem is referenced by:  funcnv3  5417  fncnv  5421