MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tposco Structured version   Visualization version   GIF version

Theorem tposco 7428
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 5692 . 2 ((𝐹𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) = (𝐹 ∘ (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})))
2 dftpos4 7416 . 2 tpos (𝐹𝐺) = ((𝐹𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
3 dftpos4 7416 . . 3 tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
43coeq2i 5315 . 2 (𝐹 ∘ tpos 𝐺) = (𝐹 ∘ (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})))
51, 2, 43eqtr4i 2683 1 tpos (𝐹𝐺) = (𝐹 ∘ tpos 𝐺)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  Vcvv 3231  cun 3605  c0 3948  {csn 4210   cuni 4468  cmpt 4762   × cxp 5141  ccnv 5142  ccom 5147  tpos ctpos 7396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-tpos 7397
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator