![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5142 | . 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 5862 | . 2 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)) |
5 | 1, 4 | bitr3i 276 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 ∘ 𝐷) ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∃wex 1781 ∈ wcel 2106 Vcvv 3473 〈cop 4628 class class class wbr 5141 ∘ ccom 5673 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-co 5678 |
This theorem is referenced by: dmcoss 5962 dmcosseq 5964 cotrgOLDOLD 6099 coiun 6244 co02 6248 coi1 6250 coass 6253 fmptco 7111 dftpos4 8212 ttrcltr 9693 fmptcof2 31751 cnvco1 34559 cnvco2 34560 txpss3v 34680 dffun10 34716 xrnss3v 37047 coiun1 42174 |
Copyright terms: Public domain | W3C validator |