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Mirrors > Home > MPE Home > Th. List > oftpos | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
oftpos | ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → tpos (𝐹 ∘𝑓 𝑅𝐺) = (tpos 𝐹 ∘𝑓 𝑅tpos 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3212 | . . . 4 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐹 ∈ V) |
3 | elex 3212 | . . . 4 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V) | |
4 | 3 | adantl 482 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ V) |
5 | funmpt 5926 | . . . 4 ⊢ Fun (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}) | |
6 | 5 | a1i 11 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → Fun (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) |
7 | dftpos4 7371 | . . . 4 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) | |
8 | tposexg 7366 | . . . . 5 ⊢ (𝐹 ∈ 𝑉 → tpos 𝐹 ∈ V) | |
9 | 8 | adantr 481 | . . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → tpos 𝐹 ∈ V) |
10 | 7, 9 | syl5eqelr 2706 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) ∈ V) |
11 | dftpos4 7371 | . . . 4 ⊢ tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) | |
12 | tposexg 7366 | . . . . 5 ⊢ (𝐺 ∈ 𝑊 → tpos 𝐺 ∈ V) | |
13 | 12 | adantl 482 | . . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → tpos 𝐺 ∈ V) |
14 | 11, 13 | syl5eqelr 2706 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) ∈ V) |
15 | ofco2 20257 | . . 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 1333 | . 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 7371 | . 2 ⊢ tpos (𝐹 ∘𝑓 𝑅𝐺) = ((𝐹 ∘𝑓 𝑅𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) | |
18 | 7, 11 | oveq12i 6662 | . 2 ⊢ (tpos 𝐹 ∘𝑓 𝑅tpos 𝐺) = ((𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) ∘𝑓 𝑅(𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))) |
19 | 16, 17, 18 | 3eqtr4g 2681 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → tpos (𝐹 ∘𝑓 𝑅𝐺) = (tpos 𝐹 ∘𝑓 𝑅tpos 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∪ cun 3572 ∅c0 3915 {csn 4177 ∪ cuni 4436 ↦ cmpt 4729 × cxp 5112 ◡ccnv 5113 ∘ ccom 5118 Fun wfun 5882 (class class class)co 6650 ∘𝑓 cof 6895 tpos ctpos 7351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-tpos 7352 |
This theorem is referenced by: mattposvs 20261 |
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