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Theorem tposco 8192
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 6221 . 2 ((𝐹𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) = (𝐹 ∘ (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})))
2 dftpos4 8180 . 2 tpos (𝐹𝐺) = ((𝐹𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
3 dftpos4 8180 . . 3 tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
43coeq2i 5820 . 2 (𝐹 ∘ tpos 𝐺) = (𝐹 ∘ (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})))
51, 2, 43eqtr4i 2771 1 tpos (𝐹𝐺) = (𝐹 ∘ tpos 𝐺)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  Vcvv 3447  cun 3912  c0 4286  {csn 4590   cuni 4869  cmpt 5192   × cxp 5635  ccnv 5636  ccom 5641  tpos ctpos 8160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-fv 6508  df-tpos 8161
This theorem is referenced by: (None)
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