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Theorem tposfun 6239
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 5236 . . 3 Fun (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})
2 funco 5238 . . 3 ((Fun 𝐹 ∧ Fun (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) → Fun (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})))
31, 2mpan2 423 . 2 (Fun 𝐹 → Fun (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})))
4 df-tpos 6224 . . 3 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
54funeqi 5219 . 2 (Fun tpos 𝐹 ↔ Fun (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})))
63, 5sylibr 133 1 (Fun 𝐹 → Fun tpos 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  cun 3119  c0 3414  {csn 3583   cuni 3796  cmpt 4050  ccnv 4610  dom cdm 4611  ccom 4615  Fun wfun 5192  tpos ctpos 6223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-fun 5200  df-tpos 6224
This theorem is referenced by:  tposfn2  6245
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