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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.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
Ref | Expression |
---|---|
rnco | ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3401 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | vex 3401 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | brco 5540 | . . . . 5 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
4 | 3 | exbii 1892 | . . . 4 ⊢ (∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑥∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
5 | excom 2155 | . . . 4 ⊢ (∃𝑥∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) | |
6 | vex 3401 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
7 | 6 | elrn 5614 | . . . . . . 7 ⊢ (𝑧 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑧) |
8 | 7 | anbi1i 617 | . . . . . 6 ⊢ ((𝑧 ∈ ran 𝐵 ∧ 𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
9 | 2 | brresi 5653 | . . . . . 6 ⊢ (𝑧(𝐴 ↾ ran 𝐵)𝑦 ↔ (𝑧 ∈ ran 𝐵 ∧ 𝑧𝐴𝑦)) |
10 | 19.41v 1992 | . . . . . 6 ⊢ (∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) | |
11 | 8, 9, 10 | 3bitr4ri 296 | . . . . 5 ⊢ (∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ 𝑧(𝐴 ↾ ran 𝐵)𝑦) |
12 | 11 | exbii 1892 | . . . 4 ⊢ (∃𝑧∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦) |
13 | 4, 5, 12 | 3bitri 289 | . . 3 ⊢ (∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦) |
14 | 2 | elrn 5614 | . . 3 ⊢ (𝑦 ∈ ran (𝐴 ∘ 𝐵) ↔ ∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦) |
15 | 2 | elrn 5614 | . . 3 ⊢ (𝑦 ∈ ran (𝐴 ↾ ran 𝐵) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦) |
16 | 13, 14, 15 | 3bitr4i 295 | . 2 ⊢ (𝑦 ∈ ran (𝐴 ∘ 𝐵) ↔ 𝑦 ∈ ran (𝐴 ↾ ran 𝐵)) |
17 | 16 | eqriv 2775 | 1 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 = wceq 1601 ∃wex 1823 ∈ wcel 2107 class class class wbr 4888 ran crn 5358 ↾ cres 5359 ∘ ccom 5361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4889 df-opab 4951 df-xp 5363 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 |
This theorem is referenced by: rnco2 5898 coeq0 5900 cofunexg 7411 1stcof 7477 2ndcof 7478 smobeth 9745 elmsubrn 32032 ftc1anclem3 34121 |
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