Theorem f1funfun 5263

Description: Two ways to express that a set A 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 set.mm contributors, 13-Aug-2004.) (Revised by Scott Fenton, 18-Apr-2021.)

Assertion

Ref	Expression
f1funfun	⊢ (A:dom A–1-1→V ↔ (Fun ◡A ∧ Fun A))

Proof of Theorem f1funfun

Step	Hyp	Ref	Expression
1		df-f1 4792	. 2 ⊢ (A:dom A–1-1→V ↔ (A:dom A–→V ∧ Fun ◡A))
2		ancom 437	. 2 ⊢ ((A:dom A–→V ∧ Fun ◡A) ↔ (Fun ◡A ∧ A:dom A–→V))
3		ssv 3291	. . . . 5 ⊢ ran A ⊆ V
4		df-f 4791	. . . . 5 ⊢ (A:dom A–→V ↔ (A Fn dom A ∧ ran A ⊆ V))
5	3, 4	mpbiran2 885	. . . 4 ⊢ (A:dom A–→V ↔ A Fn dom A)
6		funfn 5136	. . . 4 ⊢ (Fun A ↔ A Fn dom A)
7	5, 6	bitr4i 243	. . 3 ⊢ (A:dom A–→V ↔ Fun A)
8	7	anbi2i 675	. 2 ⊢ ((Fun ◡A ∧ A:dom A–→V) ↔ (Fun ◡A ∧ Fun A))
9	1, 2, 8	3bitri 262	1 ⊢ (A:dom A–1-1→V ↔ (Fun ◡A ∧ Fun A))

Syntax hints:   ↔ wb 176   ∧ wa 358  Vcvv 2859   ⊆ wss 3257  ^◡ccnv 4771  dom cdm 4772  ran crn 4773  Fun wfun 4775   Fn wfn 4776  –→wf 4777  –1-1→wf1 4778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-fn 4790  df-f 4791  df-f1 4792

This theorem is referenced by: (None)