Theorem dff1o2 6785

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 6505	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
2		df-f1 6503	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
3		df-fo 6504	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
4	2, 3	anbi12i 629	. . 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 1096	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ (Fun ◡𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)))
7	6	anbi1i 625	. . . . 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 3980	. . . . . . . 8 ⊢ (ran 𝐹 = 𝐵 → ran 𝐹 ⊆ 𝐵)
9		df-f 6502	. . . . . . . . 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 594	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴⟶𝐵)
12	11	3adant2 1132	. . . . . 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 1087   = wceq 1542   ⊆ wss 3889  ^◡ccnv 5630  ran crn 5632  Fun wfun 6492   Fn wfn 6493  ⟶wf 6494  –1-1→wf1 6495  –onto→wfo 6496  –1-1-onto→wf1o 6497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-ex 1782  df-cleq 2728  df-ss 3906  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505

This theorem is referenced by:  dff1o3  6786  dff1o4  6788  f1orn  6790  f1oi  6818  tz7.49c  8385  fiint  9237  symgfixelsi  19410  dfrelog  26529  adj1o  31965  fresf1o  32704  f1mptrn  32708  esumc  34195  sticksstones3  42587  aks6d1c6lem5  42616  cantnf2  43753  ntrneinex  44504  stoweidlem39  46467