Theorem f1cnvcnv 6587

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 6363	. 2 ⊢ (◡◡𝐴:dom 𝐴–1-1→V ↔ (◡◡𝐴:dom 𝐴⟶V ∧ Fun ◡◡◡𝐴))
2		dffn2 6519	. . . 4 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ ◡◡𝐴:dom 𝐴⟶V)
3		dmcnvcnv 5806	. . . . 5 ⊢ dom ◡◡𝐴 = dom 𝐴
4		df-fn 6361	. . . . 5 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ (Fun ◡◡𝐴 ∧ dom ◡◡𝐴 = dom 𝐴))
5	3, 4	mpbiran2 708	. . . 4 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ Fun ◡◡𝐴)
6	2, 5	bitr3i 279	. . 3 ⊢ (◡◡𝐴:dom 𝐴⟶V ↔ Fun ◡◡𝐴)
7		relcnv 5970	. . . . 5 ⊢ Rel ◡𝐴
8		dfrel2 6049	. . . . 5 ⊢ (Rel ◡𝐴 ↔ ◡◡◡𝐴 = ◡𝐴)
9	7, 8	mpbi 232	. . . 4 ⊢ ◡◡◡𝐴 = ◡𝐴
10	9	funeqi 6379	. . 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 277	1 ⊢ (◡◡𝐴:dom 𝐴–1-1→V ↔ (Fun ◡𝐴 ∧ Fun ◡◡𝐴))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1536  Vcvv 3497  ^◡ccnv 5557  dom cdm 5558  Rel wrel 5563  Fun wfun 6352   Fn wfn 6353  ⟶wf 6354  –1-1→wf1 6355

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363

This theorem is referenced by: (None)