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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 3203 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
2 | vex 3203 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | brelrn 5356 | . . . . . 6 ⊢ (𝑧𝐵𝑦 → 𝑦 ∈ ran 𝐵) |
4 | ssel 3597 | . . . . . 6 ⊢ (ran 𝐵 ⊆ 𝐶 → (𝑦 ∈ ran 𝐵 → 𝑦 ∈ 𝐶)) | |
5 | vex 3203 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
6 | 5 | brres 5402 | . . . . . . 7 ⊢ (𝑦(𝐴 ↾ 𝐶)𝑥 ↔ (𝑦𝐴𝑥 ∧ 𝑦 ∈ 𝐶)) |
7 | 6 | rbaib 947 | . . . . . 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 671 | . . . 4 ⊢ (ran 𝐵 ⊆ 𝐶 → ((𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥) ↔ (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))) |
10 | 9 | exbidv 1850 | . . 3 ⊢ (ran 𝐵 ⊆ 𝐶 → (∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥) ↔ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))) |
11 | 10 | opabbidv 4716 | . 2 ⊢ (ran 𝐵 ⊆ 𝐶 → {〈𝑧, 𝑥〉 ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥)} = {〈𝑧, 𝑥〉 ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)}) |
12 | df-co 5123 | . 2 ⊢ ((𝐴 ↾ 𝐶) ∘ 𝐵) = {〈𝑧, 𝑥〉 ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥)} | |
13 | df-co 5123 | . 2 ⊢ (𝐴 ∘ 𝐵) = {〈𝑧, 𝑥〉 ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)} | |
14 | 11, 12, 13 | 3eqtr4g 2681 | 1 ⊢ (ran 𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ∘ 𝐵) = (𝐴 ∘ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ⊆ wss 3574 class class class wbr 4653 {copab 4712 ran crn 5115 ↾ cres 5116 ∘ ccom 5118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 |
This theorem is referenced by: cocnvcnv1 5646 cores2 5648 relcoi2 5663 fco2 6059 fcoi2 6079 domss2 8119 canthp1lem2 9475 imasdsval2 16176 frmdss2 17400 gsumval3lem1 18306 gsumzres 18310 gsumzaddlem 18321 dprdf1 18432 kgencn2 21360 tsmsf1o 21948 lgamcvg2 24781 hhssims 28132 eulerpartgbij 30434 cvmlift2lem9a 31285 poimirlem9 33418 fourierdlem53 40376 funresfunco 41205 |
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