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Theorem tposfun 5797
 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 4864 . . 3 Fun (x (dom 𝐹 ∪ {∅}) ↦ {x})
2 funco 4866 . . 3 ((Fun 𝐹 Fun (x (dom 𝐹 ∪ {∅}) ↦ {x})) → Fun (𝐹 ∘ (x (dom 𝐹 ∪ {∅}) ↦ {x})))
31, 2mpan2 403 . 2 (Fun 𝐹 → Fun (𝐹 ∘ (x (dom 𝐹 ∪ {∅}) ↦ {x})))
4 df-tpos 5782 . . 3 tpos 𝐹 = (𝐹 ∘ (x (dom 𝐹 ∪ {∅}) ↦ {x}))
54funeqi 4848 . 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 2892  ∅c0 3201  {csn 3350  ∪ cuni 3554   ↦ cmpt 3792  ◡ccnv 4271  dom cdm 4272   ∘ ccom 4276  Fun wfun 4823  tpos ctpos 5781 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-fun 4831  df-tpos 5782 This theorem is referenced by:  tposfn2  5803
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