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Mirrors > Home > MPE Home > Th. List > resco | Structured version Visualization version GIF version |
Description: Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.) |
Ref | Expression |
---|---|
resco | ⊢ ((𝐴 ∘ 𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵 ↾ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 5882 | . 2 ⊢ Rel ((𝐴 ∘ 𝐵) ↾ 𝐶) | |
2 | relco 6097 | . 2 ⊢ Rel (𝐴 ∘ (𝐵 ↾ 𝐶)) | |
3 | vex 3497 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | vex 3497 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | brco 5741 | . . . . 5 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
6 | 5 | anbi2i 624 | . . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑥(𝐴 ∘ 𝐵)𝑦) ↔ (𝑥 ∈ 𝐶 ∧ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))) |
7 | 19.42v 1954 | . . . 4 ⊢ (∃𝑧(𝑥 ∈ 𝐶 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) ↔ (𝑥 ∈ 𝐶 ∧ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))) | |
8 | vex 3497 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
9 | 8 | brresi 5862 | . . . . . . 7 ⊢ (𝑥(𝐵 ↾ 𝐶)𝑧 ↔ (𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝑧)) |
10 | 9 | anbi1i 625 | . . . . . 6 ⊢ ((𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝑧) ∧ 𝑧𝐴𝑦)) |
11 | anass 471 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝑧) ∧ 𝑧𝐴𝑦) ↔ (𝑥 ∈ 𝐶 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))) | |
12 | 10, 11 | bitr2i 278 | . . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) ↔ (𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)) |
13 | 12 | exbii 1848 | . . . 4 ⊢ (∃𝑧(𝑥 ∈ 𝐶 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)) |
14 | 6, 7, 13 | 3bitr2i 301 | . . 3 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑥(𝐴 ∘ 𝐵)𝑦) ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)) |
15 | 4 | brresi 5862 | . . 3 ⊢ (𝑥((𝐴 ∘ 𝐵) ↾ 𝐶)𝑦 ↔ (𝑥 ∈ 𝐶 ∧ 𝑥(𝐴 ∘ 𝐵)𝑦)) |
16 | 3, 4 | brco 5741 | . . 3 ⊢ (𝑥(𝐴 ∘ (𝐵 ↾ 𝐶))𝑦 ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)) |
17 | 14, 15, 16 | 3bitr4i 305 | . 2 ⊢ (𝑥((𝐴 ∘ 𝐵) ↾ 𝐶)𝑦 ↔ 𝑥(𝐴 ∘ (𝐵 ↾ 𝐶))𝑦) |
18 | 1, 2, 17 | eqbrriv 5664 | 1 ⊢ ((𝐴 ∘ 𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵 ↾ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 class class class wbr 5066 ↾ cres 5557 ∘ ccom 5559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-co 5564 df-res 5567 |
This theorem is referenced by: cocnvcnv2 6111 coires1 6117 dftpos2 7909 canthp1lem2 10075 o1res 14917 gsumzaddlem 19041 tsmsf1o 22753 tsmsmhm 22754 mbfres 24245 hhssims 29051 symgcom 30727 cycpmconjslem1 30796 cycpmconjslem2 30797 erdsze2lem2 32451 cvmlift2lem9a 32550 mbfresfi 34953 cocnv 35015 xrnres 35665 xrnres2 35666 xrnres3 35667 diophrw 39376 eldioph2 39379 mbfres2cn 42263 funcoressn 43297 |
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