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

Theorem tposco 6304
Description: Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
tposco tpos (𝐹𝐺) = (𝐹 ∘ tpos 𝐺)

Proof of Theorem tposco
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 coass 5168 . 2 ((𝐹𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) = (𝐹 ∘ (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})))
2 dftpos4 6292 . 2 tpos (𝐹𝐺) = ((𝐹𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
3 dftpos4 6292 . . 3 tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
43coeq2i 4808 . 2 (𝐹 ∘ tpos 𝐺) = (𝐹 ∘ (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})))
51, 2, 43eqtr4i 2220 1 tpos (𝐹𝐺) = (𝐹 ∘ tpos 𝐺)
Colors of variables: wff set class
Syntax hints:   = wceq 1364  Vcvv 2752  cun 3142  c0 3437  {csn 3610   cuni 3827  cmpt 4082   × cxp 4645  ccnv 4646  ccom 4651  tpos ctpos 6273
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-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-fv 5246  df-tpos 6274
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator