Theorem funcnv 6594

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 6593 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 3461	. . . . . . 7 ⊢ 𝑥 ∈ V
2		vex 3461	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	brelrn 5923	. . . . . 6 ⊢ (𝑥𝐴𝑦 → 𝑦 ∈ ran 𝐴)
4	3	pm4.71ri 569	. . . . 5 ⊢ (𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
5	4	mobii 2578	. . . 4 ⊢ (∃*𝑥 𝑥𝐴𝑦 ↔ ∃*𝑥(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
6		moanimv 2649	. . . 4 ⊢ (∃*𝑥(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦) ↔ (𝑦 ∈ ran 𝐴 → ∃*𝑥 𝑥𝐴𝑦))
7	5, 6	bitri 278	. . 3 ⊢ (∃*𝑥 𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 → ∃*𝑥 𝑥𝐴𝑦))
8	7	albii 1842	. 2 ⊢ (∀𝑦∃*𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(𝑦 ∈ ran 𝐴 → ∃*𝑥 𝑥𝐴𝑦))
9		funcnv2 6593	. 2 ⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦)
10		df-ral 3080	. 2 ⊢ (∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(𝑦 ∈ ran 𝐴 → ∃*𝑥 𝑥𝐴𝑦))
11	8, 9, 10	3bitr4i 306	1 ⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1561   ∈ wcel 2145  ∃*wmo 2567  ∀wral 3079   class class class wbr 5105  ^◡ccnv 5651  ran crn 5653  Fun wfun 6519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-fun 6527

This theorem is referenced by:  funcnv3  6595  fncnv  6598