MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dff1o2 Structured version   Visualization version   GIF version

Theorem dff1o2 6839
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:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))

Proof of Theorem dff1o2
StepHypRef Expression
1 df-f1o 6551 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
2 df-f1 6549 . . . 4 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
3 df-fo 6550 . . . 4 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
42, 3anbi12i 628 . . 3 ((𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵) ↔ ((𝐹:𝐴𝐵 ∧ Fun 𝐹) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)))
5 anass 470 . . . 4 (((𝐹:𝐴𝐵 ∧ Fun 𝐹) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹:𝐴𝐵 ∧ (Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))))
6 3anan12 1097 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ (Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)))
76anbi1i 625 . . . . 5 (((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ∧ 𝐹:𝐴𝐵) ↔ ((Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ∧ 𝐹:𝐴𝐵))
8 eqimss 4041 . . . . . . . 8 (ran 𝐹 = 𝐵 → ran 𝐹𝐵)
9 df-f 6548 . . . . . . . . 9 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
109biimpri 227 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) → 𝐹:𝐴𝐵)
118, 10sylan2 594 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴𝐵)
12113adant2 1132 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴𝐵)
1312pm4.71i 561 . . . . 5 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ∧ 𝐹:𝐴𝐵))
14 ancom 462 . . . . 5 ((𝐹:𝐴𝐵 ∧ (Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))) ↔ ((Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ∧ 𝐹:𝐴𝐵))
157, 13, 143bitr4ri 304 . . . 4 ((𝐹:𝐴𝐵 ∧ (Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))) ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
165, 15bitri 275 . . 3 (((𝐹:𝐴𝐵 ∧ Fun 𝐹) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
174, 16bitri 275 . 2 ((𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵) ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
181, 17bitri 275 1 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  w3a 1088   = wceq 1542  wss 3949  ccnv 5676  ran crn 5678  Fun wfun 6538   Fn wfn 6539  wf 6540  1-1wf1 6541  ontowfo 6542  1-1-ontowf1o 6543
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551
This theorem is referenced by:  dff1o3  6840  dff1o4  6842  f1orn  6844  tz7.49c  8446  fiint  9324  symgfixelsi  19303  dfrelog  26074  adj1o  31147  fresf1o  31855  f1mptrn  31859  esumc  33049  sticksstones3  40964  cantnf2  42075  ntrneinex  42828  stoweidlem39  44755
  Copyright terms: Public domain W3C validator