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Theorem oftpos 20989
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 3510 . . . 4 (𝐹𝑉𝐹 ∈ V)
21adantr 481 . . 3 ((𝐹𝑉𝐺𝑊) → 𝐹 ∈ V)
3 elex 3510 . . . 4 (𝐺𝑊𝐺 ∈ V)
43adantl 482 . . 3 ((𝐹𝑉𝐺𝑊) → 𝐺 ∈ V)
5 funmpt 6386 . . . 4 Fun (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})
65a1i 11 . . 3 ((𝐹𝑉𝐺𝑊) → Fun (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
7 dftpos4 7900 . . . 4 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
8 tposexg 7895 . . . . 5 (𝐹𝑉 → tpos 𝐹 ∈ V)
98adantr 481 . . . 4 ((𝐹𝑉𝐺𝑊) → tpos 𝐹 ∈ V)
107, 9eqeltrrid 2915 . . 3 ((𝐹𝑉𝐺𝑊) → (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ∈ V)
11 dftpos4 7900 . . . 4 tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
12 tposexg 7895 . . . . 5 (𝐺𝑊 → tpos 𝐺 ∈ V)
1312adantl 482 . . . 4 ((𝐹𝑉𝐺𝑊) → tpos 𝐺 ∈ V)
1411, 13eqeltrrid 2915 . . 3 ((𝐹𝑉𝐺𝑊) → (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ∈ V)
15 ofco2 20988 . . 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 1369 . 2 ((𝐹𝑉𝐺𝑊) → ((𝐹f 𝑅𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) = ((𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ∘f 𝑅(𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))))
17 dftpos4 7900 . 2 tpos (𝐹f 𝑅𝐺) = ((𝐹f 𝑅𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))
187, 11oveq12i 7157 . 2 (tpos 𝐹f 𝑅tpos 𝐺) = ((𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})) ∘f 𝑅(𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥})))
1916, 17, 183eqtr4g 2878 1 ((𝐹𝑉𝐺𝑊) → tpos (𝐹f 𝑅𝐺) = (tpos 𝐹f 𝑅tpos 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cun 3931  c0 4288  {csn 4557   cuni 4830  cmpt 5137   × cxp 5546  ccnv 5547  ccom 5552  Fun wfun 6342  (class class class)co 7145  f cof 7396  tpos ctpos 7880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-tpos 7881
This theorem is referenced by:  mattposvs  20992
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