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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 5567 | . 2 ⊢ Rel ((𝐴 ∘ 𝐵) ↾ 𝐶) | |
2 | relco 5777 | . 2 ⊢ Rel (𝐴 ∘ (𝐵 ↾ 𝐶)) | |
3 | vex 3354 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | vex 3354 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | brco 5431 | . . . . 5 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
6 | 5 | anbi1i 610 | . . . 4 ⊢ ((𝑥(𝐴 ∘ 𝐵)𝑦 ∧ 𝑥 ∈ 𝐶) ↔ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶)) |
7 | 19.41v 2029 | . . . 4 ⊢ (∃𝑧((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶) ↔ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶)) | |
8 | an32 625 | . . . . . 6 ⊢ (((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶) ↔ ((𝑥𝐵𝑧 ∧ 𝑥 ∈ 𝐶) ∧ 𝑧𝐴𝑦)) | |
9 | vex 3354 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
10 | 9 | brres 5543 | . . . . . . 7 ⊢ (𝑥(𝐵 ↾ 𝐶)𝑧 ↔ (𝑥𝐵𝑧 ∧ 𝑥 ∈ 𝐶)) |
11 | 10 | anbi1i 610 | . . . . . 6 ⊢ ((𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧 ∧ 𝑥 ∈ 𝐶) ∧ 𝑧𝐴𝑦)) |
12 | 8, 11 | bitr4i 267 | . . . . 5 ⊢ (((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶) ↔ (𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)) |
13 | 12 | exbii 1924 | . . . 4 ⊢ (∃𝑧((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∧ 𝑥 ∈ 𝐶) ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)) |
14 | 6, 7, 13 | 3bitr2i 288 | . . 3 ⊢ ((𝑥(𝐴 ∘ 𝐵)𝑦 ∧ 𝑥 ∈ 𝐶) ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)) |
15 | 4 | brres 5543 | . . 3 ⊢ (𝑥((𝐴 ∘ 𝐵) ↾ 𝐶)𝑦 ↔ (𝑥(𝐴 ∘ 𝐵)𝑦 ∧ 𝑥 ∈ 𝐶)) |
16 | 3, 4 | brco 5431 | . . 3 ⊢ (𝑥(𝐴 ∘ (𝐵 ↾ 𝐶))𝑦 ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)) |
17 | 14, 15, 16 | 3bitr4i 292 | . 2 ⊢ (𝑥((𝐴 ∘ 𝐵) ↾ 𝐶)𝑦 ↔ 𝑥(𝐴 ∘ (𝐵 ↾ 𝐶))𝑦) |
18 | 1, 2, 17 | eqbrriv 5355 | 1 ⊢ ((𝐴 ∘ 𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵 ↾ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 382 = wceq 1631 ∃wex 1852 ∈ wcel 2145 class class class wbr 4786 ↾ cres 5251 ∘ ccom 5253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-br 4787 df-opab 4847 df-xp 5255 df-rel 5256 df-co 5258 df-res 5261 |
This theorem is referenced by: cocnvcnv2 5791 coires1 5797 dftpos2 7521 canthp1lem2 9677 o1res 14499 gsumzaddlem 18528 tsmsf1o 22168 tsmsmhm 22169 mbfres 23631 hhssims 28472 erdsze2lem2 31524 cvmlift2lem9a 31623 mbfresfi 33788 cocnv 33852 xrnres 34502 xrnres2 34503 xrnres3 34504 diophrw 37848 eldioph2 37851 mbfres2cn 40691 funcoressn 41727 |
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