Theorem f1f1orn 2810

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

Assertion

Ref	Expression
f1f1orn	⊢ (F:A–1-1→B → F:A–1-1-onto→ran F)

Proof of Theorem f1f1orn

Step	Hyp	Ref	Expression
1		f1f 2781	. . . 4 ⊢ (F:A–1-1→B → F:A–→B)
2		ffn 2752	. . . 4 ⊢ (F:A–→B → F Fn A)
3	1, 2	syl 12	. . 3 ⊢ (F:A–1-1→B → F Fn A)
4		df-f1 2435	. . . 4 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ Fun ◡F))
5	4	pm3.27bd 263	. . 3 ⊢ (F:A–1-1→B → Fun ◡F)
6	3, 5	jca 236	. 2 ⊢ (F:A–1-1→B → (F Fn A ∧ Fun ◡F))
7		f1orn 2809	. 2 ⊢ (F:A–1-1-onto→ran F ↔ (F Fn A ∧ Fun ◡F))
8	6, 7	sylibr 175	1 ⊢ (F:A–1-1→B → F:A–1-1-onto→ran F)

Syntax hints:   → wi 2   ∧ wa 196  ^◡ccnv 2409  ran crn 2411  Fun wfun 2416   Fn wfn 2417  –→wf 2418  –1-1→wf1 2419  –1-1-onto→wf1o 2421

This theorem is referenced by:  f1dmex 2819

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

metamath.org