Theorem f1orn 2809

Description: A one-to-one function maps onto its range.

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		df-3an 583	. 2 ⊢ ((F Fn A ∧ Fun ◡F ∧ ran F = ran F) ↔ ((F Fn A ∧ Fun ◡F) ∧ ran F = ran F))
2		f1o2 2804	. 2 ⊢ (F:A–1-1-onto→ran F ↔ (F Fn A ∧ Fun ◡F ∧ ran F = ran F))
3		cleqid 1102	. . 3 ⊢ ran F = ran F
4	3	biantru 543	. 2 ⊢ ((F Fn A ∧ Fun ◡F) ↔ ((F Fn A ∧ Fun ◡F) ∧ ran F = ran F))
5	1, 2, 4	3bitr4 158	1 ⊢ (F:A–1-1-onto→ran F ↔ (F Fn A ∧ Fun ◡F))

Syntax hints:   ↔ wb 127   ∧ wa 196   ∧ w3a 581   = wceq 1091  ^◡ccnv 2409  ran crn 2411  Fun wfun 2416   Fn wfn 2417  –1-1-onto→wf1o 2421

This theorem is referenced by:  f1f1orn 2810

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

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