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Theorem tposfun 7907
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 6392 . . 3 Fun (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})
2 funco 6394 . . 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 7891 . . 3 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
54funeqi 6375 . 2 (Fun tpos 𝐹 ↔ Fun (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})))
63, 5sylibr 236 1 (Fun 𝐹 → Fun tpos 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  cun 3933  c0 4290  {csn 4566   cuni 4837  cmpt 5145  ccnv 5553  dom cdm 5554  ccom 5558  Fun wfun 6348  tpos ctpos 7890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-fun 6356  df-tpos 7891
This theorem is referenced by:  tposfn2  7913
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