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Theorem tposfun 7650
 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 6173 . . 3 Fun (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})
2 funco 6175 . . 3 ((Fun 𝐹 ∧ Fun (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) → Fun (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})))
31, 2mpan2 681 . 2 (Fun 𝐹 → Fun (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})))
4 df-tpos 7634 . . 3 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
54funeqi 6156 . 2 (Fun tpos 𝐹 ↔ Fun (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})))
63, 5sylibr 226 1 (Fun 𝐹 → Fun tpos 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∪ cun 3790  ∅c0 4141  {csn 4398  ∪ cuni 4671   ↦ cmpt 4965  ◡ccnv 5354  dom cdm 5355   ∘ ccom 5359  Fun wfun 6129  tpos ctpos 7633 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-fun 6137  df-tpos 7634 This theorem is referenced by:  tposfn2  7656
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