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Mirrors > Home > ILE Home > Th. List > rnco | GIF version |
Description: The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.) |
Ref | Expression |
---|---|
rnco | ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2644 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | vex 2644 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | brco 4648 | . . . . 5 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
4 | 3 | exbii 1552 | . . . 4 ⊢ (∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑥∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
5 | excom 1610 | . . . 4 ⊢ (∃𝑥∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) | |
6 | ancom 264 | . . . . . . 7 ⊢ ((∃𝑥 𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑧𝐴𝑦 ∧ ∃𝑥 𝑥𝐵𝑧)) | |
7 | 19.41v 1841 | . . . . . . 7 ⊢ (∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) | |
8 | vex 2644 | . . . . . . . . 9 ⊢ 𝑧 ∈ V | |
9 | 8 | elrn 4720 | . . . . . . . 8 ⊢ (𝑧 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑧) |
10 | 9 | anbi2i 448 | . . . . . . 7 ⊢ ((𝑧𝐴𝑦 ∧ 𝑧 ∈ ran 𝐵) ↔ (𝑧𝐴𝑦 ∧ ∃𝑥 𝑥𝐵𝑧)) |
11 | 6, 7, 10 | 3bitr4i 211 | . . . . . 6 ⊢ (∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑧𝐴𝑦 ∧ 𝑧 ∈ ran 𝐵)) |
12 | 2 | brres 4761 | . . . . . 6 ⊢ (𝑧(𝐴 ↾ ran 𝐵)𝑦 ↔ (𝑧𝐴𝑦 ∧ 𝑧 ∈ ran 𝐵)) |
13 | 11, 12 | bitr4i 186 | . . . . 5 ⊢ (∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ 𝑧(𝐴 ↾ ran 𝐵)𝑦) |
14 | 13 | exbii 1552 | . . . 4 ⊢ (∃𝑧∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦) |
15 | 4, 5, 14 | 3bitri 205 | . . 3 ⊢ (∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦) |
16 | 2 | elrn 4720 | . . 3 ⊢ (𝑦 ∈ ran (𝐴 ∘ 𝐵) ↔ ∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦) |
17 | 2 | elrn 4720 | . . 3 ⊢ (𝑦 ∈ ran (𝐴 ↾ ran 𝐵) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦) |
18 | 15, 16, 17 | 3bitr4i 211 | . 2 ⊢ (𝑦 ∈ ran (𝐴 ∘ 𝐵) ↔ 𝑦 ∈ ran (𝐴 ↾ ran 𝐵)) |
19 | 18 | eqriv 2097 | 1 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1299 ∃wex 1436 ∈ wcel 1448 class class class wbr 3875 ran crn 4478 ↾ cres 4479 ∘ ccom 4481 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-br 3876 df-opab 3930 df-xp 4483 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 |
This theorem is referenced by: rnco2 4982 cofunexg 5940 1stcof 5992 2ndcof 5993 djudom 6893 |
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