Theorem dff1o2 6768

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 6488	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
2		df-f1 6486	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
3		df-fo 6487	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
4	2, 3	anbi12i 628	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵) ↔ ((𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)))
5		anass 468	. . . 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 3993	. . . . . . . 8 ⊢ (ran 𝐹 = 𝐵 → ran 𝐹 ⊆ 𝐵)
9		df-f 6485	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
10	9	biimpri 228	. . . . . . . 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 1131	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴⟶𝐵)
13	12	pm4.71i 559	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ∧ 𝐹:𝐴⟶𝐵))
14		ancom 460	. . . . 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 275	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵))
17	4, 16	bitri 275	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵) ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵))
18	1, 17	bitri 275	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ⊆ wss 3902  ^◡ccnv 5615  ran crn 5617  Fun wfun 6475   Fn wfn 6476  ⟶wf 6477  –1-1→wf1 6478  –onto→wfo 6479  –1-1-onto→wf1o 6480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ex 1781  df-cleq 2723  df-ss 3919  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488

This theorem is referenced by:  dff1o3  6769  dff1o4  6771  f1orn  6773  tz7.49c  8365  fiint  9211  symgfixelsi  19345  dfrelog  26499  adj1o  31869  fresf1o  32608  f1mptrn  32612  esumc  34059  sticksstones3  42180  aks6d1c6lem5  42209  cantnf2  43357  ntrneinex  44109  stoweidlem39  46076