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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 5701 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∃wex 1781 ∈ wcel 2111 Vcvv 3441 class class class wbr 5030 ∘ ccom 5523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-co 5528 |
This theorem is referenced by: opelco 5706 cnvco 5720 resco 6070 imaco 6071 rnco 6072 coass 6085 dffv2 6733 foeqcnvco 7034 f1eqcocnv 7035 f1eqcocnvOLD 7036 rtrclreclem3 14411 imasleval 16806 ustuqtop4 22850 metustexhalf 23163 dftr6 33099 coep 33100 coepr 33101 dfpo2 33104 brtxp 33454 pprodss4v 33458 brpprod 33459 sscoid 33487 elfuns 33489 brimg 33511 brapply 33512 brcup 33513 brcap 33514 brsuccf 33515 funpartlem 33516 brrestrict 33523 dfrecs2 33524 dfrdg4 33525 cnvssco 40306 |
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