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Mirrors > Home > ILE Home > Th. List > brcog | 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 3808 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦𝐷𝑥 ↔ 𝐴𝐷𝑥)) | |
2 | breq2 3809 | . . . 4 ⊢ (𝑧 = 𝐵 → (𝑥𝐶𝑧 ↔ 𝑥𝐶𝐵)) | |
3 | 1, 2 | bi2anan9 571 | . . 3 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → ((𝑦𝐷𝑥 ∧ 𝑥𝐶𝑧) ↔ (𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))) |
4 | 3 | exbidv 1748 | . 2 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → (∃𝑥(𝑦𝐷𝑥 ∧ 𝑥𝐶𝑧) ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))) |
5 | df-co 4400 | . 2 ⊢ (𝐶 ∘ 𝐷) = {〈𝑦, 𝑧〉 ∣ ∃𝑥(𝑦𝐷𝑥 ∧ 𝑥𝐶𝑧)} | |
6 | 4, 5 | brabga 4047 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1285 ∃wex 1422 ∈ wcel 1434 class class class wbr 3805 ∘ ccom 4395 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3916 ax-pow 3968 ax-pr 3992 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-v 2612 df-un 2986 df-in 2988 df-ss 2995 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-br 3806 df-opab 3860 df-co 4400 |
This theorem is referenced by: opelco2g 4551 brcogw 4552 brco 4554 brcodir 4762 foeqcnvco 5481 brtpos2 5920 ertr 6208 |
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