Theorem oftpos 22395

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	⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → tpos (𝐹 ∘f 𝑅𝐺) = (tpos 𝐹 ∘f 𝑅tpos 𝐺))

Proof of Theorem oftpos

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3485	. . . 4 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2	1	adantr 480	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐹 ∈ V)
3		elex 3485	. . . 4 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
4	3	adantl 481	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ V)
5		funmpt 6579	. . . 4 ⊢ Fun (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})
6	5	a1i 11	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → Fun (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))
7		dftpos4 8249	. . . 4 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))
8		tposexg 8244	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → tpos 𝐹 ∈ V)
9	8	adantr 480	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → tpos 𝐹 ∈ V)
10	7, 9	eqeltrrid 2840	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) ∈ V)
11		dftpos4 8249	. . . 4 ⊢ tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))
12		tposexg 8244	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → tpos 𝐺 ∈ V)
13	12	adantl 481	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → tpos 𝐺 ∈ V)
14	11, 13	eqeltrrid 2840	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) ∈ V)
15		ofco2 22394	. . 3 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}) ∧ (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) ∈ V ∧ (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) ∈ V)) → ((𝐹 ∘f 𝑅𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) = ((𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) ∘f 𝑅(𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))))
16	2, 4, 6, 10, 14, 15	syl23anc 1379	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘f 𝑅𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) = ((𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) ∘f 𝑅(𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))))
17		dftpos4 8249	. 2 ⊢ tpos (𝐹 ∘f 𝑅𝐺) = ((𝐹 ∘f 𝑅𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))
18	7, 11	oveq12i 7422	. 2 ⊢ (tpos 𝐹 ∘f 𝑅tpos 𝐺) = ((𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) ∘f 𝑅(𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})))
19	16, 17, 18	3eqtr4g 2796	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → tpos (𝐹 ∘f 𝑅𝐺) = (tpos 𝐹 ∘f 𝑅tpos 𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3464   ∪ cun 3929  ∅c0 4313  {csn 4606  ∪ cuni 4888   ↦ cmpt 5206   × cxp 5657  ^◡ccnv 5658   ∘ ccom 5663  Fun wfun 6530  (class class class)co 7410   ∘_f cof 7674  tpos ctpos 8229

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7676  df-tpos 8230

This theorem is referenced by:  mattposvs  22398