Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  tposssxp GIF version

Theorem tposssxp 5805
 Description: The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.)
Assertion
Ref Expression
tposssxp tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹)

Proof of Theorem tposssxp
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 df-tpos 5801 . . 3 tpos 𝐹 = (𝐹 ∘ (x (dom 𝐹 ∪ {∅}) ↦ {x}))
2 cossxp 4786 . . 3 (𝐹 ∘ (x (dom 𝐹 ∪ {∅}) ↦ {x})) ⊆ (dom (x (dom 𝐹 ∪ {∅}) ↦ {x}) × ran 𝐹)
31, 2eqsstri 2969 . 2 tpos 𝐹 ⊆ (dom (x (dom 𝐹 ∪ {∅}) ↦ {x}) × ran 𝐹)
4 eqid 2037 . . . 4 (x (dom 𝐹 ∪ {∅}) ↦ {x}) = (x (dom 𝐹 ∪ {∅}) ↦ {x})
54dmmptss 4760 . . 3 dom (x (dom 𝐹 ∪ {∅}) ↦ {x}) ⊆ (dom 𝐹 ∪ {∅})
6 xpss1 4391 . . 3 (dom (x (dom 𝐹 ∪ {∅}) ↦ {x}) ⊆ (dom 𝐹 ∪ {∅}) → (dom (x (dom 𝐹 ∪ {∅}) ↦ {x}) × ran 𝐹) ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹))
75, 6ax-mp 7 . 2 (dom (x (dom 𝐹 ∪ {∅}) ↦ {x}) × ran 𝐹) ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹)
83, 7sstri 2948 1 tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹)
 Colors of variables: wff set class Syntax hints:   ∪ cun 2909   ⊆ wss 2911  ∅c0 3218  {csn 3367  ∪ cuni 3571   ↦ cmpt 3809   × cxp 4286  ◡ccnv 4287  dom cdm 4288  ran crn 4289   ∘ ccom 4292  tpos ctpos 5800 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-mpt 3811  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-tpos 5801 This theorem is referenced by:  reltpos  5806  tposexg  5814
 Copyright terms: Public domain W3C validator