Theorem f1o2 2804

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

Assertion

Ref	Expression
f1o2	⊢ (F:A–1-1-onto→B ↔ (F Fn A ∧ Fun ◡F ∧ ran F = B))

Proof of Theorem f1o2

Step	Hyp	Ref	Expression
1		df-f1 2435	. . . . . 6 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ Fun ◡F))
2	1	pm3.27bd 263	. . . . 5 ⊢ (F:A–1-1→B → Fun ◡F)
3		df-fo 2436	. . . . . 6 ⊢ (F:A–onto→B ↔ (F Fn A ∧ ran F = B))
4	3	biimp 133	. . . . 5 ⊢ (F:A–onto→B → (F Fn A ∧ ran F = B))
5	2, 4	anim12i 268	. . . 4 ⊢ ((F:A–1-1→B ∧ F:A–onto→B) → (Fun ◡F ∧ (F Fn A ∧ ran F = B)))
6		eqimss 1548	. . . . . . . . . 10 ⊢ (ran F = B → ran F ⊆ B)
7	6	anim2i 270	. . . . . . . . 9 ⊢ ((F Fn A ∧ ran F = B) → (F Fn A ∧ ran F ⊆ B))
8		df-f 2434	. . . . . . . . 9 ⊢ (F:A–→B ↔ (F Fn A ∧ ran F ⊆ B))
9	7, 8	sylibr 175	. . . . . . . 8 ⊢ ((F Fn A ∧ ran F = B) → F:A–→B)
10	9	anim1i 269	. . . . . . 7 ⊢ (((F Fn A ∧ ran F = B) ∧ Fun ◡F) → (F:A–→B ∧ Fun ◡F))
11	10, 1	sylibr 175	. . . . . 6 ⊢ (((F Fn A ∧ ran F = B) ∧ Fun ◡F) → F:A–1-1→B)
12	11	ancoms 334	. . . . 5 ⊢ ((Fun ◡F ∧ (F Fn A ∧ ran F = B)) → F:A–1-1→B)
13	3	biimpr 134	. . . . . 6 ⊢ ((F Fn A ∧ ran F = B) → F:A–onto→B)
14	13	adantl 305	. . . . 5 ⊢ ((Fun ◡F ∧ (F Fn A ∧ ran F = B)) → F:A–onto→B)
15	12, 14	jca 236	. . . 4 ⊢ ((Fun ◡F ∧ (F Fn A ∧ ran F = B)) → (F:A–1-1→B ∧ F:A–onto→B))
16	5, 15	impbi 139	. . 3 ⊢ ((F:A–1-1→B ∧ F:A–onto→B) ↔ (Fun ◡F ∧ (F Fn A ∧ ran F = B)))
17		an12 370	. . 3 ⊢ ((Fun ◡F ∧ (F Fn A ∧ ran F = B)) ↔ (F Fn A ∧ (Fun ◡F ∧ ran F = B)))
18	16, 17	bitr 151	. 2 ⊢ ((F:A–1-1→B ∧ F:A–onto→B) ↔ (F Fn A ∧ (Fun ◡F ∧ ran F = B)))
19		df-f1o 2437	. 2 ⊢ (F:A–1-1-onto→B ↔ (F:A–1-1→B ∧ F:A–onto→B))
20		3anass 585	. 2 ⊢ ((F Fn A ∧ Fun ◡F ∧ ran F = B) ↔ (F Fn A ∧ (Fun ◡F ∧ ran F = B)))
21	18, 19, 20	3bitr4 158	1 ⊢ (F:A–1-1-onto→B ↔ (F Fn A ∧ Fun ◡F ∧ ran F = B))

Syntax hints:   ↔ wb 127   ∧ wa 196   ∧ w3a 581   = 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:  f1o4 2807  f1orn 2809  f1ocnv 2811  f1oco 2816  tz7.49c 2998  fiint 3445  infxpidmlem4 4936  infxpidmlem7 4939

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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