Theorem f1cnvcnv 5152

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 4957	. 2 ⊢ (◡◡𝐴:dom 𝐴–1-1→V ↔ (◡◡𝐴:dom 𝐴⟶V ∧ Fun ◡◡◡𝐴))
2		dffn2 5098	. . . 4 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ ◡◡𝐴:dom 𝐴⟶V)
3		dmcnvcnv 4606	. . . . 5 ⊢ dom ◡◡𝐴 = dom 𝐴
4		df-fn 4955	. . . . 5 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ (Fun ◡◡𝐴 ∧ dom ◡◡𝐴 = dom 𝐴))
5	3, 4	mpbiran2 883	. . . 4 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ Fun ◡◡𝐴)
6	2, 5	bitr3i 184	. . 3 ⊢ (◡◡𝐴:dom 𝐴⟶V ↔ Fun ◡◡𝐴)
7		relcnv 4753	. . . . 5 ⊢ Rel ◡𝐴
8		dfrel2 4821	. . . . 5 ⊢ (Rel ◡𝐴 ↔ ◡◡◡𝐴 = ◡𝐴)
9	7, 8	mpbi 143	. . . 4 ⊢ ◡◡◡𝐴 = ◡𝐴
10	9	funeqi 4972	. . 3 ⊢ (Fun ◡◡◡𝐴 ↔ Fun ◡𝐴)
11	6, 10	anbi12ci 449	. 2 ⊢ ((◡◡𝐴:dom 𝐴⟶V ∧ Fun ◡◡◡𝐴) ↔ (Fun ◡𝐴 ∧ Fun ◡◡𝐴))
12	1, 11	bitri 182	1 ⊢ (◡◡𝐴:dom 𝐴–1-1→V ↔ (Fun ◡𝐴 ∧ Fun ◡◡𝐴))

Syntax hints:   ∧ wa 102   ↔ wb 103   = wceq 1285  Vcvv 2610  ^◡ccnv 4390  dom cdm 4391  Rel wrel 4396  Fun wfun 4946   Fn wfn 4947  ⟶wf 4948  –1-1→wf1 4949

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957

This theorem is referenced by: (None)