| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > cores | Structured version Visualization version GIF version | ||
| Description: Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
| Ref | Expression |
|---|---|
| cores | ⊢ (ran 𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ∘ 𝐵) = (𝐴 ∘ 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vex 3484 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
| 2 | vex 3484 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
| 3 | 1, 2 | brelrn 5953 | . . . . . 6 ⊢ (𝑧𝐵𝑦 → 𝑦 ∈ ran 𝐵) |
| 4 | ssel 3977 | . . . . . 6 ⊢ (ran 𝐵 ⊆ 𝐶 → (𝑦 ∈ ran 𝐵 → 𝑦 ∈ 𝐶)) | |
| 5 | vex 3484 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
| 6 | 5 | brresi 6006 | . . . . . . 7 ⊢ (𝑦(𝐴 ↾ 𝐶)𝑥 ↔ (𝑦 ∈ 𝐶 ∧ 𝑦𝐴𝑥)) |
| 7 | 6 | baib 535 | . . . . . 6 ⊢ (𝑦 ∈ 𝐶 → (𝑦(𝐴 ↾ 𝐶)𝑥 ↔ 𝑦𝐴𝑥)) |
| 8 | 3, 4, 7 | syl56 36 | . . . . 5 ⊢ (ran 𝐵 ⊆ 𝐶 → (𝑧𝐵𝑦 → (𝑦(𝐴 ↾ 𝐶)𝑥 ↔ 𝑦𝐴𝑥))) |
| 9 | 8 | pm5.32d 577 | . . . 4 ⊢ (ran 𝐵 ⊆ 𝐶 → ((𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥) ↔ (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))) |
| 10 | 9 | exbidv 1921 | . . 3 ⊢ (ran 𝐵 ⊆ 𝐶 → (∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥) ↔ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))) |
| 11 | 10 | opabbidv 5209 | . 2 ⊢ (ran 𝐵 ⊆ 𝐶 → {〈𝑧, 𝑥〉 ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥)} = {〈𝑧, 𝑥〉 ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)}) |
| 12 | df-co 5694 | . 2 ⊢ ((𝐴 ↾ 𝐶) ∘ 𝐵) = {〈𝑧, 𝑥〉 ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥)} | |
| 13 | df-co 5694 | . 2 ⊢ (𝐴 ∘ 𝐵) = {〈𝑧, 𝑥〉 ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)} | |
| 14 | 11, 12, 13 | 3eqtr4g 2802 | 1 ⊢ (ran 𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ∘ 𝐵) = (𝐴 ∘ 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ⊆ wss 3951 class class class wbr 5143 {copab 5205 ran crn 5686 ↾ 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-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 |
| This theorem is referenced by: cocnvcnv1 6277 cores2 6279 relcoi2 6297 funresfunco 6607 fco2 6762 fcoi2 6783 f1ocoima 7323 domss2 9176 cottrcl 9759 canthp1lem2 10693 imasdsval2 17561 frmdss2 18876 gsumval3lem1 19923 gsumzres 19927 gsumzaddlem 19939 dprdf1 20053 kgencn2 23565 tsmsf1o 24153 lgamcvg2 27098 hhssims 31293 ccatws1f1olast 32937 symgcom 33103 cycpmconjslem1 33174 cycpmconjslem2 33175 eulerpartgbij 34374 cvmlift2lem9a 35308 poimirlem9 37636 fourierdlem53 46174 tposres3 48781 |
| Copyright terms: Public domain | W3C validator |