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Mirrors > Home > MPE Home > Th. List > opelco | Structured version Visualization version GIF version |
Description: Ordered pair membership in a composition. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.) |
Ref | Expression |
---|---|
opelco.1 | ⊢ 𝐴 ∈ V |
opelco.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
opelco | ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 ∘ 𝐷) ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5031 | . 2 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝐶 ∘ 𝐷)) | |
2 | opelco.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | opelco.2 | . . 3 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | brco 5705 | . 2 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)) |
5 | 1, 4 | bitr3i 280 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 ∘ 𝐷) ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∃wex 1781 ∈ wcel 2111 Vcvv 3441 〈cop 4531 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: dmcoss 5807 dmcosseq 5809 cotrg 5938 coiun 6076 co02 6080 coi1 6082 coass 6085 fmptco 6868 dftpos4 7894 fmptcof2 30420 cnvco1 33108 cnvco2 33109 txpss3v 33452 dffun10 33488 xrnss3v 35784 coiun1 40353 |
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