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

Theorem tposfun 5875
Description: The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
tposfun (Fun 𝐹 → Fun tpos 𝐹)

Proof of Theorem tposfun
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 funmpt 4938 . . 3 Fun (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})
2 funco 4940 . . 3 ((Fun 𝐹 ∧ Fun (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) → Fun (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})))
31, 2mpan2 401 . 2 (Fun 𝐹 → Fun (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})))
4 df-tpos 5860 . . 3 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
54funeqi 4922 . 2 (Fun tpos 𝐹 ↔ Fun (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})))
63, 5sylibr 137 1 (Fun 𝐹 → Fun tpos 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  cun 2915  c0 3224  {csn 3375   cuni 3580  cmpt 3818  ccnv 4344  dom cdm 4345  ccom 4349  Fun wfun 4896  tpos ctpos 5859
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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-fun 4904  df-tpos 5860
This theorem is referenced by:  tposfn2  5881
  Copyright terms: Public domain W3C validator