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Theorem oftpos 22474
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 3499 . . . 4 (𝐹𝑉𝐹 ∈ V)
21adantr 480 . . 3 ((𝐹𝑉𝐺𝑊) → 𝐹 ∈ V)
3 elex 3499 . . . 4 (𝐺𝑊𝐺 ∈ V)
43adantl 481 . . 3 ((𝐹𝑉𝐺𝑊) → 𝐺 ∈ V)
5 funmpt 6606 . . . 4 Fun (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})
65a1i 11 . . 3 ((𝐹𝑉𝐺𝑊) → Fun (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
7 dftpos4 8269 . . . 4 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
8 tposexg 8264 . . . . 5 (𝐹𝑉 → tpos 𝐹 ∈ V)
98adantr 480 . . . 4 ((𝐹𝑉𝐺𝑊) → tpos 𝐹 ∈ V)
107, 9eqeltrrid 2844 . . 3 ((𝐹𝑉𝐺𝑊) → (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ∈ V)
11 dftpos4 8269 . . . 4 tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
12 tposexg 8264 . . . . 5 (𝐺𝑊 → tpos 𝐺 ∈ V)
1312adantl 481 . . . 4 ((𝐹𝑉𝐺𝑊) → tpos 𝐺 ∈ V)
1411, 13eqeltrrid 2844 . . 3 ((𝐹𝑉𝐺𝑊) → (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ∈ V)
15 ofco2 22473 . . 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 1376 . 2 ((𝐹𝑉𝐺𝑊) → ((𝐹f 𝑅𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) = ((𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ∘f 𝑅(𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))))
17 dftpos4 8269 . 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 2800 1 ((𝐹𝑉𝐺𝑊) → tpos (𝐹f 𝑅𝐺) = (tpos 𝐹f 𝑅tpos 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cun 3961  c0 4339  {csn 4631   cuni 4912  cmpt 5231   × cxp 5687  ccnv 5688  ccom 5693  Fun wfun 6557  (class class class)co 7431  f cof 7695  tpos ctpos 8249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-tpos 8250
This theorem is referenced by:  mattposvs  22477
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