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Mirrors > Home > MPE Home > Th. List > brco | Structured version Visualization version GIF version |
Description: Binary relation on a composition. (Contributed by NM, 21-Sep-2004.) (Revised by Mario Carneiro, 24-Feb-2015.) |
Ref | Expression |
---|---|
opelco.1 | ⊢ 𝐴 ∈ V |
opelco.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
brco | ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelco.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | opelco.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | brcog 5732 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∃wex 1776 ∈ wcel 2110 Vcvv 3495 class class class wbr 5059 ∘ ccom 5554 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-co 5559 |
This theorem is referenced by: opelco 5737 cnvco 5751 resco 6098 imaco 6099 rnco 6100 coass 6113 dffv2 6751 foeqcnvco 7050 f1eqcocnv 7051 rtrclreclem3 14413 imasleval 16808 ustuqtop4 22847 metustexhalf 23160 dftr6 32981 coep 32982 coepr 32983 dfpo2 32986 brtxp 33336 pprodss4v 33340 brpprod 33341 sscoid 33369 elfuns 33371 brimg 33393 brapply 33394 brcup 33395 brcap 33396 brsuccf 33397 funpartlem 33398 brrestrict 33405 dfrecs2 33406 dfrdg4 33407 cnvssco 39959 |
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