Theorem brcog 4771

Description: Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)

Assertion

Ref	Expression
brcog	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)))

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.

Step	Hyp	Ref	Expression
1		breq1 3985	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦𝐷𝑥 ↔ 𝐴𝐷𝑥))
2		breq2 3986	. . . 4 ⊢ (𝑧 = 𝐵 → (𝑥𝐶𝑧 ↔ 𝑥𝐶𝐵))
3	1, 2	bi2anan9 596	. . 3 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → ((𝑦𝐷𝑥 ∧ 𝑥𝐶𝑧) ↔ (𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)))
4	3	exbidv 1813	. 2 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → (∃𝑥(𝑦𝐷𝑥 ∧ 𝑥𝐶𝑧) ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)))
5		df-co 4613	. 2 ⊢ (𝐶 ∘ 𝐷) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦𝐷𝑥 ∧ 𝑥𝐶𝑧)}
6	4, 5	brabga 4242	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343  ∃wex 1480   ∈ wcel 2136   class class class wbr 3982   ∘ ccom 4608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-co 4613

This theorem is referenced by:  opelco2g  4772  brcogw  4773  brco  4775  brcodir  4991  foeqcnvco  5758  brtpos2  6219  ertr  6516