Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > rnco | Structured version Visualization version 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 3203 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | vex 3203 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | brco 5292 | . . . . 5 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
4 | 3 | exbii 1774 | . . . 4 ⊢ (∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑥∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
5 | excom 2042 | . . . 4 ⊢ (∃𝑥∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) | |
6 | ancom 466 | . . . . . . 7 ⊢ ((∃𝑥 𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑧𝐴𝑦 ∧ ∃𝑥 𝑥𝐵𝑧)) | |
7 | 19.41v 1914 | . . . . . . 7 ⊢ (∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) | |
8 | vex 3203 | . . . . . . . . 9 ⊢ 𝑧 ∈ V | |
9 | 8 | elrn 5366 | . . . . . . . 8 ⊢ (𝑧 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑧) |
10 | 9 | anbi2i 730 | . . . . . . 7 ⊢ ((𝑧𝐴𝑦 ∧ 𝑧 ∈ ran 𝐵) ↔ (𝑧𝐴𝑦 ∧ ∃𝑥 𝑥𝐵𝑧)) |
11 | 6, 7, 10 | 3bitr4i 292 | . . . . . 6 ⊢ (∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑧𝐴𝑦 ∧ 𝑧 ∈ ran 𝐵)) |
12 | 2 | brres 5402 | . . . . . 6 ⊢ (𝑧(𝐴 ↾ ran 𝐵)𝑦 ↔ (𝑧𝐴𝑦 ∧ 𝑧 ∈ ran 𝐵)) |
13 | 11, 12 | bitr4i 267 | . . . . 5 ⊢ (∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ 𝑧(𝐴 ↾ ran 𝐵)𝑦) |
14 | 13 | exbii 1774 | . . . 4 ⊢ (∃𝑧∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦) |
15 | 4, 5, 14 | 3bitri 286 | . . 3 ⊢ (∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦) |
16 | 2 | elrn 5366 | . . 3 ⊢ (𝑦 ∈ ran (𝐴 ∘ 𝐵) ↔ ∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦) |
17 | 2 | elrn 5366 | . . 3 ⊢ (𝑦 ∈ ran (𝐴 ↾ ran 𝐵) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦) |
18 | 15, 16, 17 | 3bitr4i 292 | . 2 ⊢ (𝑦 ∈ ran (𝐴 ∘ 𝐵) ↔ 𝑦 ∈ ran (𝐴 ↾ ran 𝐵)) |
19 | 18 | eqriv 2619 | 1 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 class class class wbr 4653 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: rnco2 5642 coeq0 5644 cofunexg 7130 1stcof 7196 2ndcof 7197 smobeth 9408 elmsubrn 31425 ftc1anclem3 33487 |
Copyright terms: Public domain | W3C validator |