Theorem tposfun 8272

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.

Step	Hyp	Ref	Expression
1		funmpt 6609	. . 3 ⊢ Fun (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
2		funco 6611	. . 3 ⊢ ((Fun 𝐹 ∧ Fun (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) → Fun (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})))
3	1, 2	mpan2 691	. 2 ⊢ (Fun 𝐹 → Fun (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})))
4		df-tpos 8256	. . 3 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
5	4	funeqi 6592	. 2 ⊢ (Fun tpos 𝐹 ↔ Fun (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})))
6	3, 5	sylibr 234	1 ⊢ (Fun 𝐹 → Fun tpos 𝐹)

Syntax hints:   → wi 4   ∪ cun 3962  ∅c0 4340  {csn 4632  ∪ cuni 4913   ↦ cmpt 5232  ^◡ccnv 5689  dom cdm 5690   ∘ ccom 5694  Fun wfun 6560  tpos ctpos 8255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5303  ax-nul 5313  ax-pr 5439

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-nf 1782  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3435  df-v 3481  df-dif 3967  df-un 3969  df-ss 3981  df-nul 4341  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5584  df-xp 5696  df-rel 5697  df-cnv 5698  df-co 5699  df-fun 6568  df-tpos 8256

This theorem is referenced by:  tposfn2  8278