Theorem f1o5 2808

Description: Alternate definition of one-to-one onto function.

Assertion

Ref	Expression
f1o5	⊢ (F:A–1-1-onto→B ↔ (F:A–1-1→B ∧ ran F = B))

Proof of Theorem f1o5

Step	Hyp	Ref	Expression
1		df-f1o 2437	. 2 ⊢ (F:A–1-1-onto→B ↔ (F:A–1-1→B ∧ F:A–onto→B))
2		df-fo 2436	. . 3 ⊢ (F:A–onto→B ↔ (F Fn A ∧ ran F = B))
3	2	anbi2i 367	. 2 ⊢ ((F:A–1-1→B ∧ F:A–onto→B) ↔ (F:A–1-1→B ∧ (F Fn A ∧ ran F = B)))
4		an12 370	. . 3 ⊢ ((F:A–1-1→B ∧ (F Fn A ∧ ran F = B)) ↔ (F Fn A ∧ (F:A–1-1→B ∧ ran F = B)))
5		f1f 2781	. . . . . 6 ⊢ (F:A–1-1→B → F:A–→B)
6		ffn 2752	. . . . . 6 ⊢ (F:A–→B → F Fn A)
7	5, 6	syl 12	. . . . 5 ⊢ (F:A–1-1→B → F Fn A)
8	7	adantr 306	. . . 4 ⊢ ((F:A–1-1→B ∧ ran F = B) → F Fn A)
9	8	pm4.71ri 484	. . 3 ⊢ ((F:A–1-1→B ∧ ran F = B) ↔ (F Fn A ∧ (F:A–1-1→B ∧ ran F = B)))
10	4, 9	bitr4 154	. 2 ⊢ ((F:A–1-1→B ∧ (F Fn A ∧ ran F = B)) ↔ (F:A–1-1→B ∧ ran F = B))
11	1, 3, 10	3bitr 155	1 ⊢ (F:A–1-1-onto→B ↔ (F:A–1-1→B ∧ ran F = B))

Syntax hints:   ↔ wb 127   ∧ wa 196   = wceq 1091  ran crn 2411   Fn wfn 2417  –→wf 2418  –1-1→wf1 2419  –onto→wfo 2420  –1-1-onto→wf1o 2421

This theorem is referenced by:  mapenlem2 3385  om2uzf1o 4656

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

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