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Theorem oftpos 22441
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 3484 . . . 4 (𝐹𝑉𝐹 ∈ V)
21adantr 479 . . 3 ((𝐹𝑉𝐺𝑊) → 𝐹 ∈ V)
3 elex 3484 . . . 4 (𝐺𝑊𝐺 ∈ V)
43adantl 480 . . 3 ((𝐹𝑉𝐺𝑊) → 𝐺 ∈ V)
5 funmpt 6588 . . . 4 Fun (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})
65a1i 11 . . 3 ((𝐹𝑉𝐺𝑊) → Fun (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
7 dftpos4 8251 . . . 4 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
8 tposexg 8246 . . . . 5 (𝐹𝑉 → tpos 𝐹 ∈ V)
98adantr 479 . . . 4 ((𝐹𝑉𝐺𝑊) → tpos 𝐹 ∈ V)
107, 9eqeltrrid 2831 . . 3 ((𝐹𝑉𝐺𝑊) → (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ∈ V)
11 dftpos4 8251 . . . 4 tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
12 tposexg 8246 . . . . 5 (𝐺𝑊 → tpos 𝐺 ∈ V)
1312adantl 480 . . . 4 ((𝐹𝑉𝐺𝑊) → tpos 𝐺 ∈ V)
1411, 13eqeltrrid 2831 . . 3 ((𝐹𝑉𝐺𝑊) → (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ∈ V)
15 ofco2 22440 . . 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 1374 . 2 ((𝐹𝑉𝐺𝑊) → ((𝐹f 𝑅𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) = ((𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ∘f 𝑅(𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))))
17 dftpos4 8251 . 2 tpos (𝐹f 𝑅𝐺) = ((𝐹f 𝑅𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
187, 11oveq12i 7427 . 2 (tpos 𝐹f 𝑅tpos 𝐺) = ((𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ∘f 𝑅(𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})))
1916, 17, 183eqtr4g 2791 1 ((𝐹𝑉𝐺𝑊) → tpos (𝐹f 𝑅𝐺) = (tpos 𝐹f 𝑅tpos 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  Vcvv 3464  cun 3946  c0 4324  {csn 4625   cuni 4907  cmpt 5228   × cxp 5672  ccnv 5673  ccom 5678  Fun wfun 6539  (class class class)co 7415  f cof 7679  tpos ctpos 8231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3466  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4325  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4908  df-iun 4997  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6497  df-fun 6547  df-fn 6548  df-f 6549  df-f1 6550  df-fo 6551  df-f1o 6552  df-fv 6553  df-ov 7418  df-oprab 7419  df-mpo 7420  df-of 7681  df-tpos 8232
This theorem is referenced by:  mattposvs  22444
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