Theorem f1cnvcnv 5043

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 4850	. 2 ⊢ (◡◡A:dom A–1-1→V ↔ (◡◡A:dom A⟶V ∧ Fun ◡◡◡A))
2		dffn2 4990	. . . 4 ⊢ (◡◡A Fn dom A ↔ ◡◡A:dom A⟶V)
3		dmcnvcnv 4501	. . . . 5 ⊢ dom ◡◡A = dom A
4		df-fn 4848	. . . . 5 ⊢ (◡◡A Fn dom A ↔ (Fun ◡◡A ∧ dom ◡◡A = dom A))
5	3, 4	mpbiran2 847	. . . 4 ⊢ (◡◡A Fn dom A ↔ Fun ◡◡A)
6	2, 5	bitr3i 175	. . 3 ⊢ (◡◡A:dom A⟶V ↔ Fun ◡◡A)
7		relcnv 4646	. . . . 5 ⊢ Rel ◡A
8		dfrel2 4714	. . . . 5 ⊢ (Rel ◡A ↔ ◡◡◡A = ◡A)
9	7, 8	mpbi 133	. . . 4 ⊢ ◡◡◡A = ◡A
10	9	funeqi 4865	. . 3 ⊢ (Fun ◡◡◡A ↔ Fun ◡A)
11	6, 10	anbi12ci 434	. 2 ⊢ ((◡◡A:dom A⟶V ∧ Fun ◡◡◡A) ↔ (Fun ◡A ∧ Fun ◡◡A))
12	1, 11	bitri 173	1 ⊢ (◡◡A:dom A–1-1→V ↔ (Fun ◡A ∧ Fun ◡◡A))

Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242  Vcvv 2551  ^◡ccnv 4287  dom cdm 4288  Rel wrel 4293  Fun wfun 4839   Fn wfn 4840  ⟶wf 4841  –1-1→wf1 4842

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850

This theorem is referenced by: (None)