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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 5106 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦𝐷𝑥 ↔ 𝐴𝐷𝑥)) | |
2 | breq2 5107 | . . . 4 ⊢ (𝑧 = 𝐵 → (𝑥𝐶𝑧 ↔ 𝑥𝐶𝐵)) | |
3 | 1, 2 | bi2anan9 637 | . . 3 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → ((𝑦𝐷𝑥 ∧ 𝑥𝐶𝑧) ↔ (𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))) |
4 | 3 | exbidv 1924 | . 2 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → (∃𝑥(𝑦𝐷𝑥 ∧ 𝑥𝐶𝑧) ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))) |
5 | df-co 5640 | . 2 ⊢ (𝐶 ∘ 𝐷) = {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑦𝐷𝑥 ∧ 𝑥𝐶𝑧)} | |
6 | 4, 5 | brabga 5489 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 class class class wbr 5103 ∘ ccom 5635 |
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 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
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 2714 df-cleq 2728 df-clel 2814 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5166 df-co 5640 |
This theorem is referenced by: opelco2g 5821 brcogw 5822 brco 5824 brcodir 6071 predtrss 6274 brtpos2 8155 ertr 8621 relexpindlem 14902 znleval 20908 fcoinvbr 31371 opelco3 34181 brxrn 36768 eqvreltr 37001 frege124d 41938 funressnfv 45172 dfatcolem 45382 |
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