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Theorem tposfun 5816
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 4881 . . 3 Fun (x (dom 𝐹 ∪ {∅}) ↦ {x})
2 funco 4883 . . 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 5801 . . 3 tpos 𝐹 = (𝐹 ∘ (x (dom 𝐹 ∪ {∅}) ↦ {x}))
54funeqi 4865 . 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 2909  c0 3218  {csn 3367   cuni 3571  cmpt 3809  ccnv 4287  dom cdm 4288  ccom 4292  Fun wfun 4839  tpos ctpos 5800
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-fun 4847  df-tpos 5801
This theorem is referenced by:  tposfn2  5822
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