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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 3482 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | vex 3482 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | brco 5884 | . . . . 5 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
4 | 3 | exbii 1845 | . . . 4 ⊢ (∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑥∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
5 | excom 2160 | . . . 4 ⊢ (∃𝑥∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) | |
6 | vex 3482 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
7 | 6 | elrn 5907 | . . . . . . 7 ⊢ (𝑧 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑧) |
8 | 7 | anbi1i 624 | . . . . . 6 ⊢ ((𝑧 ∈ ran 𝐵 ∧ 𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
9 | 2 | brresi 6009 | . . . . . 6 ⊢ (𝑧(𝐴 ↾ ran 𝐵)𝑦 ↔ (𝑧 ∈ ran 𝐵 ∧ 𝑧𝐴𝑦)) |
10 | 19.41v 1947 | . . . . . 6 ⊢ (∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) | |
11 | 8, 9, 10 | 3bitr4ri 304 | . . . . 5 ⊢ (∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ 𝑧(𝐴 ↾ ran 𝐵)𝑦) |
12 | 11 | exbii 1845 | . . . 4 ⊢ (∃𝑧∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦) |
13 | 4, 5, 12 | 3bitri 297 | . . 3 ⊢ (∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦) |
14 | 2 | elrn 5907 | . . 3 ⊢ (𝑦 ∈ ran (𝐴 ∘ 𝐵) ↔ ∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦) |
15 | 2 | elrn 5907 | . . 3 ⊢ (𝑦 ∈ ran (𝐴 ↾ ran 𝐵) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦) |
16 | 13, 14, 15 | 3bitr4i 303 | . 2 ⊢ (𝑦 ∈ ran (𝐴 ∘ 𝐵) ↔ 𝑦 ∈ ran (𝐴 ↾ ran 𝐵)) |
17 | 16 | eqriv 2732 | 1 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 class class class wbr 5148 ran crn 5690 ↾ cres 5691 ∘ ccom 5693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 |
This theorem is referenced by: rnco2 6275 coeq0 6277 focofo 6834 cofunexg 7972 1stcof 8043 2ndcof 8044 smobeth 10624 cycpmconjv 33145 elmsubrn 35513 ftc1anclem3 37682 |
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