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Theorem tposfun 8248
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 6592 . . 3 Fun (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})
2 funco 6594 . . 3 ((Fun 𝐹 ∧ Fun (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) → Fun (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})))
31, 2mpan2 689 . 2 (Fun 𝐹 → Fun (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})))
4 df-tpos 8232 . . 3 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
54funeqi 6575 . 2 (Fun tpos 𝐹 ↔ Fun (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})))
63, 5sylibr 233 1 (Fun 𝐹 → Fun tpos 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  cun 3942  c0 4322  {csn 4630   cuni 4909  cmpt 5232  ccnv 5677  dom cdm 5678  ccom 5682  Fun wfun 6543  tpos ctpos 8231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-fun 6551  df-tpos 8232
This theorem is referenced by:  tposfn2  8254
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