Theorem dff1o2 5056

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 4836	. 2 ⊢ (𝐹:A–1-1-onto→B ↔ (𝐹:A–1-1→B ∧ 𝐹:A–onto→B))
2		df-f1 4834	. . . 4 ⊢ (𝐹:A–1-1→B ↔ (𝐹:A⟶B ∧ Fun ◡𝐹))
3		df-fo 4835	. . . 4 ⊢ (𝐹:A–onto→B ↔ (𝐹 Fn A ∧ ran 𝐹 = B))
4	2, 3	anbi12i 436	. . 3 ⊢ ((𝐹:A–1-1→B ∧ 𝐹:A–onto→B) ↔ ((𝐹:A⟶B ∧ Fun ◡𝐹) ∧ (𝐹 Fn A ∧ ran 𝐹 = B)))
5		anass 383	. . . 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 885	. . . . . 6 ⊢ ((𝐹 Fn A ∧ Fun ◡𝐹 ∧ ran 𝐹 = B) ↔ (Fun ◡𝐹 ∧ (𝐹 Fn A ∧ ran 𝐹 = B)))
7	6	anbi1i 434	. . . . 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 2974	. . . . . . . 8 ⊢ (ran 𝐹 = B → ran 𝐹 ⊆ B)
9		df-f 4833	. . . . . . . . 9 ⊢ (𝐹:A⟶B ↔ (𝐹 Fn A ∧ ran 𝐹 ⊆ B))
10	9	biimpri 124	. . . . . . . 8 ⊢ ((𝐹 Fn A ∧ ran 𝐹 ⊆ B) → 𝐹:A⟶B)
11	8, 10	sylan2 270	. . . . . . 7 ⊢ ((𝐹 Fn A ∧ ran 𝐹 = B) → 𝐹:A⟶B)
12	11	3adant2 911	. . . . . 6 ⊢ ((𝐹 Fn A ∧ Fun ◡𝐹 ∧ ran 𝐹 = B) → 𝐹:A⟶B)
13	12	pm4.71i 371	. . . . 5 ⊢ ((𝐹 Fn A ∧ Fun ◡𝐹 ∧ ran 𝐹 = B) ↔ ((𝐹 Fn A ∧ Fun ◡𝐹 ∧ ran 𝐹 = B) ∧ 𝐹:A⟶B))
14		ancom 253	. . . . 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 202	. . . 4 ⊢ ((𝐹:A⟶B ∧ (Fun ◡𝐹 ∧ (𝐹 Fn A ∧ ran 𝐹 = B))) ↔ (𝐹 Fn A ∧ Fun ◡𝐹 ∧ ran 𝐹 = B))
16	5, 15	bitri 173	. . 3 ⊢ (((𝐹:A⟶B ∧ Fun ◡𝐹) ∧ (𝐹 Fn A ∧ ran 𝐹 = B)) ↔ (𝐹 Fn A ∧ Fun ◡𝐹 ∧ ran 𝐹 = B))
17	4, 16	bitri 173	. 2 ⊢ ((𝐹:A–1-1→B ∧ 𝐹:A–onto→B) ↔ (𝐹 Fn A ∧ Fun ◡𝐹 ∧ ran 𝐹 = B))
18	1, 17	bitri 173	1 ⊢ (𝐹:A–1-1-onto→B ↔ (𝐹 Fn A ∧ Fun ◡𝐹 ∧ ran 𝐹 = B))

Syntax hints:   ∧ wa 97   ↔ wb 98   ∧ w3a 873   = wceq 1228   ⊆ wss 2894  ^◡ccnv 4271  ran crn 4273  Fun wfun 4823   Fn wfn 4824  ⟶wf 4825  –1-1→wf1 4826  –onto→wfo 4827  –1-1-onto→wf1o 4828

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-3an 875  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-in 2901  df-ss 2908  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836

This theorem is referenced by:  dff1o3  5057  dff1o4  5059  f1orn  5061