Theorem tposfun 8167

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 6514	. . 3 ⊢ Fun (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
2		funco 6516	. . 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 8151	. . 3 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
5	4	funeqi 6497	. 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 3895  ∅c0 4278  {csn 4571  ∪ cuni 4854   ↦ cmpt 5167  ^◡ccnv 5610  dom cdm 5611   ∘ ccom 5615  Fun wfun 6470  tpos ctpos 8150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-fun 6478  df-tpos 8151

This theorem is referenced by:  tposfn2  8173