Theorem oftpos 20674

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 3414	. . . 4 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2	1	adantr 474	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐹 ∈ V)
3		elex 3414	. . . 4 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
4	3	adantl 475	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ V)
5		funmpt 6175	. . . 4 ⊢ Fun (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})
6	5	a1i 11	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → Fun (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))
7		dftpos4 7655	. . . 4 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))
8		tposexg 7650	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → tpos 𝐹 ∈ V)
9	8	adantr 474	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → tpos 𝐹 ∈ V)
10	7, 9	syl5eqelr 2864	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) ∈ V)
11		dftpos4 7655	. . . 4 ⊢ tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))
12		tposexg 7650	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → tpos 𝐺 ∈ V)
13	12	adantl 475	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → tpos 𝐺 ∈ V)
14	11, 13	syl5eqelr 2864	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) ∈ V)
15		ofco2 20673	. . 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 1445	. 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 7655	. 2 ⊢ tpos (𝐹 ∘𝑓 𝑅𝐺) = ((𝐹 ∘𝑓 𝑅𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))
18	7, 11	oveq12i 6936	. 2 ⊢ (tpos 𝐹 ∘𝑓 𝑅tpos 𝐺) = ((𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) ∘𝑓 𝑅(𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})))
19	16, 17, 18	3eqtr4g 2839	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → tpos (𝐹 ∘𝑓 𝑅𝐺) = (tpos 𝐹 ∘𝑓 𝑅tpos 𝐺))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  Vcvv 3398   ∪ cun 3790  ∅c0 4141  {csn 4398  ∪ cuni 4673   ↦ cmpt 4967   × cxp 5355  ^◡ccnv 5356   ∘ ccom 5361  Fun wfun 6131  (class class class)co 6924   ∘_𝑓 cof 7174  tpos ctpos 7635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-of 7176  df-tpos 7636

This theorem is referenced by:  mattposvs  20677