Theorem f1cnvcnv 6729

Description: Two ways to express that a set 𝐴 (not necessarily a function) is one-to-one. Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un₂ (A)" for one-to-one. We do not introduce a separate notation since we rarely use it. (Contributed by NM, 13-Aug-2004.)

Assertion

Ref	Expression
f1cnvcnv	⊢ (◡◡𝐴:dom 𝐴–1-1→V ↔ (Fun ◡𝐴 ∧ Fun ◡◡𝐴))

Proof of Theorem f1cnvcnv

Step	Hyp	Ref	Expression
1		df-f1 6487	. 2 ⊢ (◡◡𝐴:dom 𝐴–1-1→V ↔ (◡◡𝐴:dom 𝐴⟶V ∧ Fun ◡◡◡𝐴))
2		dffn2 6654	. . . 4 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ ◡◡𝐴:dom 𝐴⟶V)
3		dmcnvcnv 5875	. . . . 5 ⊢ dom ◡◡𝐴 = dom 𝐴
4		df-fn 6485	. . . . 5 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ (Fun ◡◡𝐴 ∧ dom ◡◡𝐴 = dom 𝐴))
5	3, 4	mpbiran2 710	. . . 4 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ Fun ◡◡𝐴)
6	2, 5	bitr3i 277	. . 3 ⊢ (◡◡𝐴:dom 𝐴⟶V ↔ Fun ◡◡𝐴)
7		relcnv 6055	. . . . 5 ⊢ Rel ◡𝐴
8		dfrel2 6138	. . . . 5 ⊢ (Rel ◡𝐴 ↔ ◡◡◡𝐴 = ◡𝐴)
9	7, 8	mpbi 230	. . . 4 ⊢ ◡◡◡𝐴 = ◡𝐴
10	9	funeqi 6503	. . 3 ⊢ (Fun ◡◡◡𝐴 ↔ Fun ◡𝐴)
11	6, 10	anbi12ci 629	. 2 ⊢ ((◡◡𝐴:dom 𝐴⟶V ∧ Fun ◡◡◡𝐴) ↔ (Fun ◡𝐴 ∧ Fun ◡◡𝐴))
12	1, 11	bitri 275	1 ⊢ (◡◡𝐴:dom 𝐴–1-1→V ↔ (Fun ◡𝐴 ∧ Fun ◡◡𝐴))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  Vcvv 3436  ^◡ccnv 5618  dom cdm 5619  Rel wrel 5624  Fun wfun 6476   Fn wfn 6477  ⟶wf 6478  –1-1→wf1 6479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487

This theorem is referenced by: (None)