Theorem dff1o2 6823

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 6538	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
2		df-f1 6536	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
3		df-fo 6537	. . . 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 4017	. . . . . . . 8 ⊢ (ran 𝐹 = 𝐵 → ran 𝐹 ⊆ 𝐵)
9		df-f 6535	. . . . . . . . 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 1540   ⊆ wss 3926  ^◡ccnv 5653  ran crn 5655  Fun wfun 6525   Fn wfn 6526  ⟶wf 6527  –1-1→wf1 6528  –onto→wfo 6529  –1-1-onto→wf1o 6530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ex 1780  df-cleq 2727  df-ss 3943  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538

This theorem is referenced by:  dff1o3  6824  dff1o4  6826  f1orn  6828  tz7.49c  8460  fiint  9338  fiintOLD  9339  symgfixelsi  19416  dfrelog  26526  adj1o  31875  fresf1o  32609  f1mptrn  32613  esumc  34082  sticksstones3  42161  aks6d1c6lem5  42190  cantnf2  43349  ntrneinex  44101  stoweidlem39  46068