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Theorem oftpos 22458
Description: The transposition of the value of a function operation for two functions is the value of the function operation for the two functions transposed. (Contributed by Stefan O'Rear, 17-Jul-2018.)
Assertion
Ref Expression
oftpos ((𝐹𝑉𝐺𝑊) → tpos (𝐹f 𝑅𝐺) = (tpos 𝐹f 𝑅tpos 𝐺))

Proof of Theorem oftpos
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elex 3501 . . . 4 (𝐹𝑉𝐹 ∈ V)
21adantr 480 . . 3 ((𝐹𝑉𝐺𝑊) → 𝐹 ∈ V)
3 elex 3501 . . . 4 (𝐺𝑊𝐺 ∈ V)
43adantl 481 . . 3 ((𝐹𝑉𝐺𝑊) → 𝐺 ∈ V)
5 funmpt 6604 . . . 4 Fun (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})
65a1i 11 . . 3 ((𝐹𝑉𝐺𝑊) → Fun (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
7 dftpos4 8270 . . . 4 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
8 tposexg 8265 . . . . 5 (𝐹𝑉 → tpos 𝐹 ∈ V)
98adantr 480 . . . 4 ((𝐹𝑉𝐺𝑊) → tpos 𝐹 ∈ V)
107, 9eqeltrrid 2846 . . 3 ((𝐹𝑉𝐺𝑊) → (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ∈ V)
11 dftpos4 8270 . . . 4 tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
12 tposexg 8265 . . . . 5 (𝐺𝑊 → tpos 𝐺 ∈ V)
1312adantl 481 . . . 4 ((𝐹𝑉𝐺𝑊) → tpos 𝐺 ∈ V)
1411, 13eqeltrrid 2846 . . 3 ((𝐹𝑉𝐺𝑊) → (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ∈ V)
15 ofco2 22457 . . 3 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}) ∧ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ∈ V ∧ (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ∈ V)) → ((𝐹f 𝑅𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) = ((𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ∘f 𝑅(𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))))
162, 4, 6, 10, 14, 15syl23anc 1379 . 2 ((𝐹𝑉𝐺𝑊) → ((𝐹f 𝑅𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) = ((𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ∘f 𝑅(𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))))
17 dftpos4 8270 . 2 tpos (𝐹f 𝑅𝐺) = ((𝐹f 𝑅𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
187, 11oveq12i 7443 . 2 (tpos 𝐹f 𝑅tpos 𝐺) = ((𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ∘f 𝑅(𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})))
1916, 17, 183eqtr4g 2802 1 ((𝐹𝑉𝐺𝑊) → tpos (𝐹f 𝑅𝐺) = (tpos 𝐹f 𝑅tpos 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cun 3949  c0 4333  {csn 4626   cuni 4907  cmpt 5225   × cxp 5683  ccnv 5684  ccom 5689  Fun wfun 6555  (class class class)co 7431  f cof 7695  tpos ctpos 8250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-tpos 8251
This theorem is referenced by:  mattposvs  22461
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