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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 4584 | . 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 5214 | . 2 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)) |
5 | 1, 4 | bitr3i 265 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 ∘ 𝐷) ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 〈cop 4131 class class class wbr 4583 ∘ ccom 5042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-co 5047 |
This theorem is referenced by: dmcoss 5306 dmcosseq 5308 cotrg 5426 coiun 5562 co02 5566 coi1 5568 coass 5571 fmptco 6303 dftpos4 7258 fmptcof2 28839 cnvco1 30903 cnvco2 30904 txpss3v 31155 dffun10 31191 coiun1 36963 |
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