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Theorem brcog 4529
Description: Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)
Assertion
Ref Expression
brcog ((𝐴𝑉𝐵𝑊) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷
Allowed substitution hints:   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem brcog
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 3794 . . . 4 (𝑦 = 𝐴 → (𝑦𝐷𝑥𝐴𝐷𝑥))
2 breq2 3795 . . . 4 (𝑧 = 𝐵 → (𝑥𝐶𝑧𝑥𝐶𝐵))
31, 2bi2anan9 548 . . 3 ((𝑦 = 𝐴𝑧 = 𝐵) → ((𝑦𝐷𝑥𝑥𝐶𝑧) ↔ (𝐴𝐷𝑥𝑥𝐶𝐵)))
43exbidv 1722 . 2 ((𝑦 = 𝐴𝑧 = 𝐵) → (∃𝑥(𝑦𝐷𝑥𝑥𝐶𝑧) ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
5 df-co 4381 . 2 (𝐶𝐷) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦𝐷𝑥𝑥𝐶𝑧)}
64, 5brabga 4028 1 ((𝐴𝑉𝐵𝑊) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409   class class class wbr 3791  ccom 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-co 4381
This theorem is referenced by:  opelco2g  4530  brcogw  4531  brco  4533  brcodir  4739  foeqcnvco  5457  brtpos2  5896  ertr  6151
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