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Mirrors > Home > ILE Home > Th. List > tposco | GIF version |
Description: Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
tposco | ⊢ tpos (𝐹 ∘ 𝐺) = (𝐹 ∘ tpos 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coass 5168 | . 2 ⊢ ((𝐹 ∘ 𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) = (𝐹 ∘ (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))) | |
2 | dftpos4 6292 | . 2 ⊢ tpos (𝐹 ∘ 𝐺) = ((𝐹 ∘ 𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) | |
3 | dftpos4 6292 | . . 3 ⊢ tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) | |
4 | 3 | coeq2i 4808 | . 2 ⊢ (𝐹 ∘ tpos 𝐺) = (𝐹 ∘ (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))) |
5 | 1, 2, 4 | 3eqtr4i 2220 | 1 ⊢ tpos (𝐹 ∘ 𝐺) = (𝐹 ∘ tpos 𝐺) |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 Vcvv 2752 ∪ cun 3142 ∅c0 3437 {csn 3610 ∪ cuni 3827 ↦ cmpt 4082 × cxp 4645 ◡ccnv 4646 ∘ ccom 4651 tpos ctpos 6273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 ax-un 4454 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-br 4022 df-opab 4083 df-mpt 4084 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-fv 5246 df-tpos 6274 |
This theorem is referenced by: (None) |
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