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Mirrors > Home > ILE Home > Th. List > cores | 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 2560 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
2 | vex 2560 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | brelrn 4567 | . . . . . 6 ⊢ (𝑧𝐵𝑦 → 𝑦 ∈ ran 𝐵) |
4 | ssel 2939 | . . . . . 6 ⊢ (ran 𝐵 ⊆ 𝐶 → (𝑦 ∈ ran 𝐵 → 𝑦 ∈ 𝐶)) | |
5 | vex 2560 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
6 | 5 | brres 4618 | . . . . . . 7 ⊢ (𝑦(𝐴 ↾ 𝐶)𝑥 ↔ (𝑦𝐴𝑥 ∧ 𝑦 ∈ 𝐶)) |
7 | 6 | rbaib 830 | . . . . . 6 ⊢ (𝑦 ∈ 𝐶 → (𝑦(𝐴 ↾ 𝐶)𝑥 ↔ 𝑦𝐴𝑥)) |
8 | 3, 4, 7 | syl56 30 | . . . . 5 ⊢ (ran 𝐵 ⊆ 𝐶 → (𝑧𝐵𝑦 → (𝑦(𝐴 ↾ 𝐶)𝑥 ↔ 𝑦𝐴𝑥))) |
9 | 8 | pm5.32d 423 | . . . 4 ⊢ (ran 𝐵 ⊆ 𝐶 → ((𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥) ↔ (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))) |
10 | 9 | exbidv 1706 | . . 3 ⊢ (ran 𝐵 ⊆ 𝐶 → (∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥) ↔ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))) |
11 | 10 | opabbidv 3823 | . 2 ⊢ (ran 𝐵 ⊆ 𝐶 → {〈𝑧, 𝑥〉 ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥)} = {〈𝑧, 𝑥〉 ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)}) |
12 | df-co 4354 | . 2 ⊢ ((𝐴 ↾ 𝐶) ∘ 𝐵) = {〈𝑧, 𝑥〉 ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥)} | |
13 | df-co 4354 | . 2 ⊢ (𝐴 ∘ 𝐵) = {〈𝑧, 𝑥〉 ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)} | |
14 | 11, 12, 13 | 3eqtr4g 2097 | 1 ⊢ (ran 𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ∘ 𝐵) = (𝐴 ∘ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∃wex 1381 ∈ wcel 1393 ⊆ wss 2917 class class class wbr 3764 {copab 3817 ran crn 4346 ↾ cres 4347 ∘ ccom 4349 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 |
This theorem is referenced by: cocnvcnv1 4831 cores2 4833 relcoi2 4848 fco2 5057 fcoi2 5071 |
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