Theorem oftpos 20460

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 3352	. . . 4 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2	1	adantr 472	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐹 ∈ V)
3		elex 3352	. . . 4 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
4	3	adantl 473	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ V)
5		funmpt 6087	. . . 4 ⊢ Fun (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})
6	5	a1i 11	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → Fun (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))
7		dftpos4 7540	. . . 4 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))
8		tposexg 7535	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → tpos 𝐹 ∈ V)
9	8	adantr 472	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → tpos 𝐹 ∈ V)
10	7, 9	syl5eqelr 2844	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) ∈ V)
11		dftpos4 7540	. . . 4 ⊢ tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))
12		tposexg 7535	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → tpos 𝐺 ∈ V)
13	12	adantl 473	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → tpos 𝐺 ∈ V)
14	11, 13	syl5eqelr 2844	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) ∈ V)
15		ofco2 20459	. . 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 1484	. 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 7540	. 2 ⊢ tpos (𝐹 ∘𝑓 𝑅𝐺) = ((𝐹 ∘𝑓 𝑅𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))
18	7, 11	oveq12i 6825	. 2 ⊢ (tpos 𝐹 ∘𝑓 𝑅tpos 𝐺) = ((𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) ∘𝑓 𝑅(𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})))
19	16, 17, 18	3eqtr4g 2819	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → tpos (𝐹 ∘𝑓 𝑅𝐺) = (tpos 𝐹 ∘𝑓 𝑅tpos 𝐺))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  Vcvv 3340   ∪ cun 3713  ∅c0 4058  {csn 4321  ∪ cuni 4588   ↦ cmpt 4881   × cxp 5264  ^◡ccnv 5265   ∘ ccom 5270  Fun wfun 6043  (class class class)co 6813   ∘_𝑓 cof 7060  tpos ctpos 7520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-tpos 7521

This theorem is referenced by:  mattposvs  20463