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Mirrors > Home > MPE Home > Th. List > brcog | Structured version Visualization version GIF version |
Description: Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.) |
Ref | Expression |
---|---|
brcog | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5062 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦𝐷𝑥 ↔ 𝐴𝐷𝑥)) | |
2 | breq2 5063 | . . . 4 ⊢ (𝑧 = 𝐵 → (𝑥𝐶𝑧 ↔ 𝑥𝐶𝐵)) | |
3 | 1, 2 | bi2anan9 637 | . . 3 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → ((𝑦𝐷𝑥 ∧ 𝑥𝐶𝑧) ↔ (𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))) |
4 | 3 | exbidv 1918 | . 2 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → (∃𝑥(𝑦𝐷𝑥 ∧ 𝑥𝐶𝑧) ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))) |
5 | df-co 5559 | . 2 ⊢ (𝐶 ∘ 𝐷) = {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑦𝐷𝑥 ∧ 𝑥𝐶𝑧)} | |
6 | 4, 5 | brabga 5414 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 class class class wbr 5059 ∘ ccom 5554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-co 5559 |
This theorem is referenced by: opelco2g 5733 brcogw 5734 brco 5736 brcodir 5974 brtpos2 7892 ertr 8298 relexpindlem 14416 znleval 20695 fcoinvbr 30352 opelco3 33013 brxrn 35620 eqvreltr 35836 frege124d 40099 funressnfv 43271 dfatcolem 43447 |
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