Theorem f1orn 6618

Description: A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)

Assertion

Ref	Expression
f1orn	⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹))

Proof of Theorem f1orn

Step	Hyp	Ref	Expression
1		dff1o2 6613	. 2 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = ran 𝐹))
2		eqid 2819	. . 3 ⊢ ran 𝐹 = ran 𝐹
3		df-3an 1084	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹) ∧ ran 𝐹 = ran 𝐹))
4	2, 3	mpbiran2 708	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹))
5	1, 4	bitri 277	1 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1082   = wceq 1531  ^◡ccnv 5547  ran crn 5549  Fun wfun 6342   Fn wfn 6343  –1-1-onto→wf1o 6347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-in 3941  df-ss 3950  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355

This theorem is referenced by:  f1f1orn  6619  infdifsn  9112  efopnlem2  25232  cycpmcl  30751