Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dff1o6 Structured version   Visualization version   GIF version

Theorem dff1o6 6431
 Description: A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.)
Assertion
Ref Expression
dff1o6 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem dff1o6
StepHypRef Expression
1 df-f1o 5811 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
2 dff13 6416 . . 3 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
3 df-fo 5810 . . 3 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
42, 3anbi12i 729 . 2 ((𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵) ↔ ((𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)))
5 df-3an 1033 . . 3 ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
6 eqimss 3620 . . . . . . 7 (ran 𝐹 = 𝐵 → ran 𝐹𝐵)
76anim2i 591 . . . . . 6 ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
8 df-f 5808 . . . . . 6 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
97, 8sylibr 223 . . . . 5 ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴𝐵)
109pm4.71ri 663 . . . 4 ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ↔ (𝐹:𝐴𝐵 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)))
1110anbi1i 727 . . 3 (((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ↔ ((𝐹:𝐴𝐵 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
12 an32 835 . . 3 (((𝐹:𝐴𝐵 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ↔ ((𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)))
135, 11, 123bitrri 286 . 2 (((𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
141, 4, 133bitri 285 1 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∀wral 2896   ⊆ wss 3540  ran crn 5039   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812 This theorem is referenced by:  soisores  6477  f1otrg  25551  f1otrge  25552  nbgraf1olem1  25970  nbgraf1olem5  25974  grpoinvf  26770  bra11  28351  diaf11N  35356  dibf11N  35468  lcfrlem9  35857  mapd1o  35955  hdmapf1oN  36175  hgmapf1oN  36213  rmxypairf1o  36494
 Copyright terms: Public domain W3C validator