Theorem dff1o2 5291

Description: Alternate definition of one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 10-Feb-1997.) (Revised by set.mm contributors, 22-Oct-2011.)

Assertion

Ref	Expression
dff1o2	⊢ (F:A–1-1-onto→B ↔ (F Fn A ∧ Fun ◡F ∧ ran F = B))

Proof of Theorem dff1o2

Step	Hyp	Ref	Expression
1		df-f1o 4794	. 2 ⊢ (F:A–1-1-onto→B ↔ (F:A–1-1→B ∧ F:A–onto→B))
2		df-f1 4792	. . 3 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ Fun ◡F))
3		df-fo 4793	. . 3 ⊢ (F:A–onto→B ↔ (F Fn A ∧ ran F = B))
4	2, 3	anbi12i 678	. 2 ⊢ ((F:A–1-1→B ∧ F:A–onto→B) ↔ ((F:A–→B ∧ Fun ◡F) ∧ (F Fn A ∧ ran F = B)))
5		ancom 437	. . . 4 ⊢ ((F:A–→B ∧ (Fun ◡F ∧ (F Fn A ∧ ran F = B))) ↔ ((Fun ◡F ∧ (F Fn A ∧ ran F = B)) ∧ F:A–→B))
6		3anass 938	. . . . . 6 ⊢ ((F Fn A ∧ Fun ◡F ∧ ran F = B) ↔ (F Fn A ∧ (Fun ◡F ∧ ran F = B)))
7		an12 772	. . . . . 6 ⊢ ((F Fn A ∧ (Fun ◡F ∧ ran F = B)) ↔ (Fun ◡F ∧ (F Fn A ∧ ran F = B)))
8	6, 7	bitri 240	. . . . 5 ⊢ ((F Fn A ∧ Fun ◡F ∧ ran F = B) ↔ (Fun ◡F ∧ (F Fn A ∧ ran F = B)))
9	8	anbi1i 676	. . . 4 ⊢ (((F Fn A ∧ Fun ◡F ∧ ran F = B) ∧ F:A–→B) ↔ ((Fun ◡F ∧ (F Fn A ∧ ran F = B)) ∧ F:A–→B))
10	5, 9	bitr4i 243	. . 3 ⊢ ((F:A–→B ∧ (Fun ◡F ∧ (F Fn A ∧ ran F = B))) ↔ ((F Fn A ∧ Fun ◡F ∧ ran F = B) ∧ F:A–→B))
11		anass 630	. . 3 ⊢ (((F:A–→B ∧ Fun ◡F) ∧ (F Fn A ∧ ran F = B)) ↔ (F:A–→B ∧ (Fun ◡F ∧ (F Fn A ∧ ran F = B))))
12		eqimss 3323	. . . . . 6 ⊢ (ran F = B → ran F ⊆ B)
13		df-f 4791	. . . . . . 7 ⊢ (F:A–→B ↔ (F Fn A ∧ ran F ⊆ B))
14	13	biimpri 197	. . . . . 6 ⊢ ((F Fn A ∧ ran F ⊆ B) → F:A–→B)
15	12, 14	sylan2 460	. . . . 5 ⊢ ((F Fn A ∧ ran F = B) → F:A–→B)
16	15	3adant2 974	. . . 4 ⊢ ((F Fn A ∧ Fun ◡F ∧ ran F = B) → F:A–→B)
17	16	pm4.71i 613	. . 3 ⊢ ((F Fn A ∧ Fun ◡F ∧ ran F = B) ↔ ((F Fn A ∧ Fun ◡F ∧ ran F = B) ∧ F:A–→B))
18	10, 11, 17	3bitr4i 268	. 2 ⊢ (((F:A–→B ∧ Fun ◡F) ∧ (F Fn A ∧ ran F = B)) ↔ (F Fn A ∧ Fun ◡F ∧ ran F = B))
19	1, 4, 18	3bitri 262	1 ⊢ (F:A–1-1-onto→B ↔ (F Fn A ∧ Fun ◡F ∧ ran F = B))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ⊆ wss 3257  ^◡ccnv 4771  ran crn 4773  Fun wfun 4775   Fn wfn 4776  –→wf 4777  –1-1→wf1 4778  –onto→wfo 4779  –1-1-onto→wf1o 4780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794

This theorem is referenced by:  dff1o3  5292  dff1o4  5294  f1orn  5296  fundmen  6043  enmap1  6074  enprmap  6082  sbthlem3  6205