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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 6023 | . 2 ⊢ Rel ((𝐴 ∘ 𝐵) ↾ 𝐶) | |
| 2 | relco 6126 | . 2 ⊢ Rel (𝐴 ∘ (𝐵 ↾ 𝐶)) | |
| 3 | vex 3484 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 4 | vex 3484 | . . . . . 6 ⊢ 𝑦 ∈ V | |
| 5 | 3, 4 | brco 5881 | . . . . 5 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
| 6 | 5 | anbi2i 623 | . . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑥(𝐴 ∘ 𝐵)𝑦) ↔ (𝑥 ∈ 𝐶 ∧ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))) |
| 7 | 19.42v 1953 | . . . 4 ⊢ (∃𝑧(𝑥 ∈ 𝐶 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) ↔ (𝑥 ∈ 𝐶 ∧ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))) | |
| 8 | vex 3484 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
| 9 | 8 | brresi 6006 | . . . . . . 7 ⊢ (𝑥(𝐵 ↾ 𝐶)𝑧 ↔ (𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝑧)) |
| 10 | 9 | anbi1i 624 | . . . . . 6 ⊢ ((𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝑧) ∧ 𝑧𝐴𝑦)) |
| 11 | anass 468 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝑧) ∧ 𝑧𝐴𝑦) ↔ (𝑥 ∈ 𝐶 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))) | |
| 12 | 10, 11 | bitr2i 276 | . . . . 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 299 | . . 3 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑥(𝐴 ∘ 𝐵)𝑦) ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)) |
| 15 | 4 | brresi 6006 | . . 3 ⊢ (𝑥((𝐴 ∘ 𝐵) ↾ 𝐶)𝑦 ↔ (𝑥 ∈ 𝐶 ∧ 𝑥(𝐴 ∘ 𝐵)𝑦)) |
| 16 | 3, 4 | brco 5881 | . . 3 ⊢ (𝑥(𝐴 ∘ (𝐵 ↾ 𝐶))𝑦 ↔ ∃𝑧(𝑥(𝐵 ↾ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)) |
| 17 | 14, 15, 16 | 3bitr4i 303 | . 2 ⊢ (𝑥((𝐴 ∘ 𝐵) ↾ 𝐶)𝑦 ↔ 𝑥(𝐴 ∘ (𝐵 ↾ 𝐶))𝑦) |
| 18 | 1, 2, 17 | eqbrriv 5801 | 1 ⊢ ((𝐴 ∘ 𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵 ↾ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 class class class wbr 5143 ↾ cres 5687 ∘ ccom 5689 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-co 5694 df-res 5697 |
| This theorem is referenced by: cocnvcnv2 6278 coires1 6284 dftpos2 8268 ttrclco 9758 canthp1lem2 10693 o1res 15596 gsumzaddlem 19939 tsmsf1o 24153 tsmsmhm 24154 mbfres 25679 hhssims 31293 symgcom 33103 cycpmconjslem1 33174 cycpmconjslem2 33175 erdsze2lem2 35209 cvmlift2lem9a 35308 mbfresfi 37673 cocnv 37732 xrnres 38403 xrnres2 38404 xrnres3 38405 diophrw 42770 eldioph2 42773 mbfres2cn 45973 funcoressn 47054 tposrescnv 48779 |
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