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Theorem tposco 8238
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 6255 . 2 ((𝐹𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) = (𝐹 ∘ (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})))
2 dftpos4 8226 . 2 tpos (𝐹𝐺) = ((𝐹𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
3 dftpos4 8226 . . 3 tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
43coeq2i 5851 . 2 (𝐹 ∘ tpos 𝐺) = (𝐹 ∘ (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})))
51, 2, 43eqtr4i 2762 1 tpos (𝐹𝐺) = (𝐹 ∘ tpos 𝐺)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  Vcvv 3466  cun 3939  c0 4315  {csn 4621   cuni 4900  cmpt 5222   × cxp 5665  ccnv 5666  ccom 5671  tpos ctpos 8206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-fv 6542  df-tpos 8207
This theorem is referenced by: (None)
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