Theorem dff1o2 6721

Description: Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 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 6440	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
2		df-f1 6438	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
3		df-fo 6439	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
4	2, 3	anbi12i 627	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵) ↔ ((𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)))
5		anass 469	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹:𝐴⟶𝐵 ∧ (Fun ◡𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))))
6		3anan12 1095	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ (Fun ◡𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)))
7	6	anbi1i 624	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ∧ 𝐹:𝐴⟶𝐵) ↔ ((Fun ◡𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ∧ 𝐹:𝐴⟶𝐵))
8		eqimss 3977	. . . . . . . 8 ⊢ (ran 𝐹 = 𝐵 → ran 𝐹 ⊆ 𝐵)
9		df-f 6437	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
10	9	biimpri 227	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵)
11	8, 10	sylan2 593	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴⟶𝐵)
12	11	3adant2 1130	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴⟶𝐵)
13	12	pm4.71i 560	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ∧ 𝐹:𝐴⟶𝐵))
14		ancom 461	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (Fun ◡𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))) ↔ ((Fun ◡𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ∧ 𝐹:𝐴⟶𝐵))
15	7, 13, 14	3bitr4ri 304	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (Fun ◡𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))) ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵))
16	5, 15	bitri 274	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵))
17	4, 16	bitri 274	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵) ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵))
18	1, 17	bitri 274	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ⊆ wss 3887  ^◡ccnv 5588  ran crn 5590  Fun wfun 6427   Fn wfn 6428  ⟶wf 6429  –1-1→wf1 6430  –onto→wfo 6431  –1-1-onto→wf1o 6432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440

This theorem is referenced by:  dff1o3  6722  dff1o4  6724  f1orn  6726  tz7.49c  8277  fiint  9091  symgfixelsi  19043  dfrelog  25721  adj1o  30256  fresf1o  30966  f1mptrn  30970  esumc  32019  sticksstones3  40104  ntrneinex  41687  stoweidlem39  43580