![]() |
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 3354 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | vex 3354 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | brco 5429 | . . . . 5 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
4 | 3 | exbii 1924 | . . . 4 ⊢ (∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑥∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
5 | excom 2198 | . . . 4 ⊢ (∃𝑥∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) | |
6 | ancom 448 | . . . . . . 7 ⊢ ((∃𝑥 𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑧𝐴𝑦 ∧ ∃𝑥 𝑥𝐵𝑧)) | |
7 | 19.41v 2029 | . . . . . . 7 ⊢ (∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) | |
8 | vex 3354 | . . . . . . . . 9 ⊢ 𝑧 ∈ V | |
9 | 8 | elrn 5502 | . . . . . . . 8 ⊢ (𝑧 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑧) |
10 | 9 | anbi2i 609 | . . . . . . 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 5541 | . . . . . 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 1924 | . . . 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 5502 | . . 3 ⊢ (𝑦 ∈ ran (𝐴 ∘ 𝐵) ↔ ∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦) |
17 | 2 | elrn 5502 | . . 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 2768 | 1 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 382 = wceq 1631 ∃wex 1852 ∈ wcel 2145 class class class wbr 4786 ran crn 5250 ↾ cres 5251 ∘ ccom 5253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-br 4787 df-opab 4847 df-xp 5255 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 |
This theorem is referenced by: rnco2 5784 coeq0 5786 cofunexg 7275 1stcof 7343 2ndcof 7344 smobeth 9608 elmsubrn 31756 ftc1anclem3 33812 |
Copyright terms: Public domain | W3C validator |