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

Theorem tposssxp 6302
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 6298 . . 3 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
2 cossxp 5188 . . 3 (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) × ran 𝐹)
31, 2eqsstri 3211 . 2 tpos 𝐹 ⊆ (dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) × ran 𝐹)
4 eqid 2193 . . . 4 (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) = (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})
54dmmptss 5162 . . 3 dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ⊆ (dom 𝐹 ∪ {∅})
6 xpss1 4769 . . 3 (dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ⊆ (dom 𝐹 ∪ {∅}) → (dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) × ran 𝐹) ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹))
75, 6ax-mp 5 . 2 (dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) × ran 𝐹) ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹)
83, 7sstri 3188 1 tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹)
Colors of variables: wff set class
Syntax hints:  cun 3151  wss 3153  c0 3446  {csn 3618   cuni 3835  cmpt 4090   × cxp 4657  ccnv 4658  dom cdm 4659  ran crn 4660  ccom 4663  tpos ctpos 6297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-mpt 4092  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-tpos 6298
This theorem is referenced by:  reltpos  6303  tposexg  6311
  Copyright terms: Public domain W3C validator