Theorem f1f1orn 3705

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 3671	. . . 4 ⊢ (F:A–1-1→B → F:A–→B)
2		ffn 3633	. . . 4 ⊢ (F:A–→B → F Fn A)
3	1, 2	syl 10	. . 3 ⊢ (F:A–1-1→B → F Fn A)
4		df-f1 3201	. . . 4 ⊢ (F:A–1-1→B ↔ (F:A–→B ⋀ Fun ◡F))
5	4	pm3.27bi 326	. . 3 ⊢ (F:A–1-1→B → Fun ◡F)
6	3, 5	jca 288	. 2 ⊢ (F:A–1-1→B → (F Fn A ⋀ Fun ◡F))
7		f1orn 3704	. 2 ⊢ (F:A–1-1-onto→ran F ↔ (F Fn A ⋀ Fun ◡F))
8	6, 7	sylibr 200	1 ⊢ (F:A–1-1→B → F:A–1-1-onto→ran F)

Syntax hints:   → wi 3   ⋀ wa 223  ^◡ccnv 3175  ran crn 3177  Fun wfun 3182   Fn wfn 3183  –→wf 3184  –1-1→wf1 3185  –1-1-onto→wf1o 3187

This theorem is referenced by:  f1dmex 3716

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-in 2054  df-ss 2056  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203

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