Theorem f1cnvcnv 6789

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 6545	. 2 ⊢ (◡◡𝐴:dom 𝐴–1-1→V ↔ (◡◡𝐴:dom 𝐴⟶V ∧ Fun ◡◡◡𝐴))
2		dffn2 6711	. . . 4 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ ◡◡𝐴:dom 𝐴⟶V)
3		dmcnvcnv 5927	. . . . 5 ⊢ dom ◡◡𝐴 = dom 𝐴
4		df-fn 6543	. . . . 5 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ (Fun ◡◡𝐴 ∧ dom ◡◡𝐴 = dom 𝐴))
5	3, 4	mpbiran2 722	. . . 4 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ Fun ◡◡𝐴)
6	2, 5	bitr3i 280	. . 3 ⊢ (◡◡𝐴:dom 𝐴⟶V ↔ Fun ◡◡𝐴)
7		relcnv 6110	. . . . 5 ⊢ Rel ◡𝐴
8		dfrel2 6191	. . . . 5 ⊢ (Rel ◡𝐴 ↔ ◡◡◡𝐴 = ◡𝐴)
9	7, 8	mpbi 233	. . . 4 ⊢ ◡◡◡𝐴 = ◡𝐴
10	9	funeqi 6561	. . 3 ⊢ (Fun ◡◡◡𝐴 ↔ Fun ◡𝐴)
11	6, 10	anbi12ci 640	. 2 ⊢ ((◡◡𝐴:dom 𝐴⟶V ∧ Fun ◡◡◡𝐴) ↔ (Fun ◡𝐴 ∧ Fun ◡◡𝐴))
12	1, 11	bitri 278	1 ⊢ (◡◡𝐴:dom 𝐴–1-1→V ↔ (Fun ◡𝐴 ∧ Fun ◡◡𝐴))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1568  Vcvv 3462  ^◡ccnv 5664  dom cdm 5665  Rel wrel 5670  Fun wfun 6534   Fn wfn 6535  ⟶wf 6536  –1-1→wf1 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545

This theorem is referenced by: (None)