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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 2554 | . . . . . . 7 ⊢ z ∈ V | |
2 | vex 2554 | . . . . . . 7 ⊢ y ∈ V | |
3 | 1, 2 | brelrn 4510 | . . . . . 6 ⊢ (zBy → y ∈ ran B) |
4 | ssel 2933 | . . . . . 6 ⊢ (ran B ⊆ 𝐶 → (y ∈ ran B → y ∈ 𝐶)) | |
5 | vex 2554 | . . . . . . . 8 ⊢ x ∈ V | |
6 | 5 | brres 4561 | . . . . . . 7 ⊢ (y(A ↾ 𝐶)x ↔ (yAx ∧ y ∈ 𝐶)) |
7 | 6 | rbaib 829 | . . . . . 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 1703 | . . 3 ⊢ (ran B ⊆ 𝐶 → (∃y(zBy ∧ y(A ↾ 𝐶)x) ↔ ∃y(zBy ∧ yAx))) |
11 | 10 | opabbidv 3814 | . 2 ⊢ (ran B ⊆ 𝐶 → {〈z, x〉 ∣ ∃y(zBy ∧ y(A ↾ 𝐶)x)} = {〈z, x〉 ∣ ∃y(zBy ∧ yAx)}) |
12 | df-co 4297 | . 2 ⊢ ((A ↾ 𝐶) ∘ B) = {〈z, x〉 ∣ ∃y(zBy ∧ y(A ↾ 𝐶)x)} | |
13 | df-co 4297 | . 2 ⊢ (A ∘ B) = {〈z, x〉 ∣ ∃y(zBy ∧ yAx)} | |
14 | 11, 12, 13 | 3eqtr4g 2094 | 1 ⊢ (ran B ⊆ 𝐶 → ((A ↾ 𝐶) ∘ B) = (A ∘ B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∃wex 1378 ∈ wcel 1390 ⊆ wss 2911 class class class wbr 3755 {copab 3808 ran crn 4289 ↾ cres 4290 ∘ ccom 4292 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 |
This theorem is referenced by: cocnvcnv1 4774 cores2 4776 relcoi2 4791 fco2 5000 fcoi2 5014 |
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