Theorem tposfun 8234

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 6572	. . 3 ⊢ Fun (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
2		funco 6574	. . 3 ⊢ ((Fun 𝐹 ∧ Fun (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) → Fun (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})))
3	1, 2	mpan2 703	. 2 ⊢ (Fun 𝐹 → Fun (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})))
4		df-tpos 8218	. . 3 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
5	4	funeqi 6555	. 2 ⊢ (Fun tpos 𝐹 ↔ Fun (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})))
6	3, 5	sylibr 237	1 ⊢ (Fun 𝐹 → Fun tpos 𝐹)

Syntax hints:   → wi 4   ∪ cun 3911  ∅c0 4294  {csn 4591  ∪ cuni 4873   ↦ cmpt 5193  ^◡ccnv 5658  dom cdm 5659   ∘ ccom 5663  Fun wfun 6528  tpos ctpos 8217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-fun 6536  df-tpos 8218

This theorem is referenced by:  tposfn2  8240