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Mirrors > Home > MPE Home > Th. List > tposco | Structured version Visualization version 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 6263 | . 2 ⊢ ((𝐹 ∘ 𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) = (𝐹 ∘ (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))) | |
2 | dftpos4 8244 | . 2 ⊢ tpos (𝐹 ∘ 𝐺) = ((𝐹 ∘ 𝐺) ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) | |
3 | dftpos4 8244 | . . 3 ⊢ tpos 𝐺 = (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) | |
4 | 3 | coeq2i 5857 | . 2 ⊢ (𝐹 ∘ tpos 𝐺) = (𝐹 ∘ (𝐺 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥}))) |
5 | 1, 2, 4 | 3eqtr4i 2765 | 1 ⊢ tpos (𝐹 ∘ 𝐺) = (𝐹 ∘ tpos 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 Vcvv 3469 ∪ cun 3942 ∅c0 4318 {csn 4624 ∪ cuni 4903 ↦ cmpt 5225 × cxp 5670 ◡ccnv 5671 ∘ ccom 5676 tpos ctpos 8224 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-fv 6550 df-tpos 8225 |
This theorem is referenced by: (None) |
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