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| 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 5146 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦𝐷𝑥 ↔ 𝐴𝐷𝑥)) | |
| 2 | breq2 5147 | . . . 4 ⊢ (𝑧 = 𝐵 → (𝑥𝐶𝑧 ↔ 𝑥𝐶𝐵)) | |
| 3 | 1, 2 | bi2anan9 638 | . . 3 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → ((𝑦𝐷𝑥 ∧ 𝑥𝐶𝑧) ↔ (𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))) | 
| 4 | 3 | exbidv 1921 | . 2 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → (∃𝑥(𝑦𝐷𝑥 ∧ 𝑥𝐶𝑧) ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))) | 
| 5 | df-co 5694 | . 2 ⊢ (𝐶 ∘ 𝐷) = {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑦𝐷𝑥 ∧ 𝑥𝐶𝑧)} | |
| 6 | 4, 5 | brabga 5539 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 class class class wbr 5143 ∘ ccom 5689 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-co 5694 | 
| This theorem is referenced by: opelco2g 5878 brcogw 5879 brco 5881 brcodir 6139 predtrss 6343 brtpos2 8257 ertr 8760 relexpindlem 15102 znleval 21573 fcoinvbr 32618 opelco3 35775 brxrn 38375 eqvreltr 38608 frege124d 43774 funressnfv 47055 dfatcolem 47267 | 
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