ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dffo5 GIF version

Theorem dffo5 5343
Description: Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
Assertion
Ref Expression
dffo5 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦

Proof of Theorem dffo5
StepHypRef Expression
1 dffo4 5342 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
2 rexex 2385 . . . . 5 (∃𝑥𝐴 𝑥𝐹𝑦 → ∃𝑥 𝑥𝐹𝑦)
32ralimi 2401 . . . 4 (∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦 → ∀𝑦𝐵𝑥 𝑥𝐹𝑦)
43anim2i 328 . . 3 ((𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦) → (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦))
5 ffn 5073 . . . . . . . . 9 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
6 fnbr 5028 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑥𝐹𝑦) → 𝑥𝐴)
76ex 112 . . . . . . . . 9 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦𝑥𝐴))
85, 7syl 14 . . . . . . . 8 (𝐹:𝐴𝐵 → (𝑥𝐹𝑦𝑥𝐴))
98ancrd 313 . . . . . . 7 (𝐹:𝐴𝐵 → (𝑥𝐹𝑦 → (𝑥𝐴𝑥𝐹𝑦)))
109eximdv 1776 . . . . . 6 (𝐹:𝐴𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥(𝑥𝐴𝑥𝐹𝑦)))
11 df-rex 2329 . . . . . 6 (∃𝑥𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
1210, 11syl6ibr 155 . . . . 5 (𝐹:𝐴𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥𝐴 𝑥𝐹𝑦))
1312ralimdv 2405 . . . 4 (𝐹:𝐴𝐵 → (∀𝑦𝐵𝑥 𝑥𝐹𝑦 → ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
1413imdistani 427 . . 3 ((𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦) → (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
154, 14impbii 121 . 2 ((𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦))
161, 15bitri 177 1 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wex 1397  wcel 1409  wral 2323  wrex 2324   class class class wbr 3791   Fn wfn 4924  wf 4925  ontowfo 4927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-fo 4935  df-fv 4937
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator