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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 3389 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
2 | vex 3389 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | brelrn 5561 | . . . . . 6 ⊢ (𝑧𝐵𝑦 → 𝑦 ∈ ran 𝐵) |
4 | ssel 3793 | . . . . . 6 ⊢ (ran 𝐵 ⊆ 𝐶 → (𝑦 ∈ ran 𝐵 → 𝑦 ∈ 𝐶)) | |
5 | vex 3389 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
6 | 5 | brresi 5610 | . . . . . . 7 ⊢ (𝑦(𝐴 ↾ 𝐶)𝑥 ↔ (𝑦 ∈ 𝐶 ∧ 𝑦𝐴𝑥)) |
7 | 6 | baib 532 | . . . . . 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 573 | . . . 4 ⊢ (ran 𝐵 ⊆ 𝐶 → ((𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥) ↔ (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))) |
10 | 9 | exbidv 2017 | . . 3 ⊢ (ran 𝐵 ⊆ 𝐶 → (∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥) ↔ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))) |
11 | 10 | opabbidv 4910 | . 2 ⊢ (ran 𝐵 ⊆ 𝐶 → {〈𝑧, 𝑥〉 ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥)} = {〈𝑧, 𝑥〉 ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)}) |
12 | df-co 5322 | . 2 ⊢ ((𝐴 ↾ 𝐶) ∘ 𝐵) = {〈𝑧, 𝑥〉 ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥)} | |
13 | df-co 5322 | . 2 ⊢ (𝐴 ∘ 𝐵) = {〈𝑧, 𝑥〉 ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)} | |
14 | 11, 12, 13 | 3eqtr4g 2859 | 1 ⊢ (ran 𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ∘ 𝐵) = (𝐴 ∘ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∃wex 1875 ∈ wcel 2157 ⊆ wss 3770 class class class wbr 4844 {copab 4906 ran crn 5314 ↾ cres 5315 ∘ ccom 5317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pr 5098 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3388 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-sn 4370 df-pr 4372 df-op 4376 df-br 4845 df-opab 4907 df-xp 5319 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 |
This theorem is referenced by: cocnvcnv1 5866 cores2 5868 relcoi2 5883 funresfunco 6143 fco2 6275 fcoi2 6295 domss2 8362 canthp1lem2 9764 imasdsval2 16490 frmdss2 17715 gsumval3lem1 18620 gsumzres 18624 gsumzaddlem 18635 dprdf1 18747 kgencn2 21688 tsmsf1o 22275 lgamcvg2 25132 hhssims 28656 eulerpartgbij 30949 cvmlift2lem9a 31801 poimirlem9 33906 fourierdlem53 41114 |
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