Theorem f1o3 2805

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

Assertion

Ref	Expression
f1o3	⊢ (F:A–1-1-onto→B ↔ (F:A–onto→B ∧ Fun ◡F))

Proof of Theorem f1o3

Step	Hyp	Ref	Expression
1		an23 371	. . 3 ⊢ (((F:A–→B ∧ Fun ◡F) ∧ (F Fn A ∧ ran F = B)) ↔ ((F:A–→B ∧ (F Fn A ∧ ran F = B)) ∧ Fun ◡F))
2		df-f1 2435	. . . 4 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ Fun ◡F))
3		df-fo 2436	. . . 4 ⊢ (F:A–onto→B ↔ (F Fn A ∧ ran F = B))
4	2, 3	anbi12i 369	. . 3 ⊢ ((F:A–1-1→B ∧ F:A–onto→B) ↔ ((F:A–→B ∧ Fun ◡F) ∧ (F Fn A ∧ ran F = B)))
5		eqimss 1548	. . . . . . 7 ⊢ (ran F = B → ran F ⊆ B)
6	5	anim2i 270	. . . . . 6 ⊢ ((F Fn A ∧ ran F = B) → (F Fn A ∧ ran F ⊆ B))
7		df-f 2434	. . . . . 6 ⊢ (F:A–→B ↔ (F Fn A ∧ ran F ⊆ B))
8	6, 7	sylibr 175	. . . . 5 ⊢ ((F Fn A ∧ ran F = B) → F:A–→B)
9	8	pm4.71ri 484	. . . 4 ⊢ ((F Fn A ∧ ran F = B) ↔ (F:A–→B ∧ (F Fn A ∧ ran F = B)))
10	9	anbi1i 368	. . 3 ⊢ (((F Fn A ∧ ran F = B) ∧ Fun ◡F) ↔ ((F:A–→B ∧ (F Fn A ∧ ran F = B)) ∧ Fun ◡F))
11	1, 4, 10	3bitr4 158	. 2 ⊢ ((F:A–1-1→B ∧ F:A–onto→B) ↔ ((F Fn A ∧ ran F = B) ∧ Fun ◡F))
12		df-f1o 2437	. 2 ⊢ (F:A–1-1-onto→B ↔ (F:A–1-1→B ∧ F:A–onto→B))
13	3	anbi1i 368	. 2 ⊢ ((F:A–onto→B ∧ Fun ◡F) ↔ ((F Fn A ∧ ran F = B) ∧ Fun ◡F))
14	11, 12, 13	3bitr4 158	1 ⊢ (F:A–1-1-onto→B ↔ (F:A–onto→B ∧ Fun ◡F))

Syntax hints:   ↔ wb 127   ∧ wa 196   = wceq 1091   ⊆ wss 1487  ^◡ccnv 2409  ran crn 2411  Fun wfun 2416   Fn wfn 2417  –→wf 2418  –1-1→wf1 2419  –onto→wfo 2420  –1-1-onto→wf1o 2421

This theorem is referenced by:  f1ofo 2806  f1ores 2813  f11o 2821  f1oi 2825  ssdomg 3311  mapenlem1 3384  phplem5 3407  php3 3411

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-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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