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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 5730 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))) | |
4 | 1, 2, 3 | mp2an 688 | 1 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∃wex 1771 ∈ wcel 2105 Vcvv 3492 class class class wbr 5057 ∘ ccom 5552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-co 5557 |
This theorem is referenced by: opelco 5735 cnvco 5749 resco 6096 imaco 6097 rnco 6098 coass 6111 dffv2 6749 foeqcnvco 7047 f1eqcocnv 7048 rtrclreclem3 14407 imasleval 16802 ustuqtop4 22780 metustexhalf 23093 dftr6 32883 coep 32884 coepr 32885 dfpo2 32888 brtxp 33238 pprodss4v 33242 brpprod 33243 sscoid 33271 elfuns 33273 brimg 33295 brapply 33296 brcup 33297 brcap 33298 brsuccf 33299 funpartlem 33300 brrestrict 33307 dfrecs2 33308 dfrdg4 33309 cnvssco 39844 |
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