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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 5013 | . 2 ⊢ ((𝐹 ∘ 𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) = (𝐹 ∘ (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))) | |
2 | dftpos4 6112 | . 2 ⊢ tpos (𝐹 ∘ 𝐺) = ((𝐹 ∘ 𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) | |
3 | dftpos4 6112 | . . 3 ⊢ tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) | |
4 | 3 | coeq2i 4657 | . 2 ⊢ (𝐹 ∘ tpos 𝐺) = (𝐹 ∘ (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))) |
5 | 1, 2, 4 | 3eqtr4i 2143 | 1 ⊢ tpos (𝐹 ∘ 𝐺) = (𝐹 ∘ tpos 𝐺) |
Colors of variables: wff set class |
Syntax hints: = wceq 1312 Vcvv 2655 ∪ cun 3033 ∅c0 3327 {csn 3491 ∪ cuni 3700 ↦ cmpt 3947 × cxp 4495 ◡ccnv 4496 ∘ ccom 4501 tpos ctpos 6093 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-rab 2397 df-v 2657 df-sbc 2877 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-fv 5087 df-tpos 6094 |
This theorem is referenced by: (None) |
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