Theorem reloprab 5890

Description: An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.)

Assertion

Ref	Expression
reloprab	⊢ Rel {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem reloprab

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfoprab2 5889	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2	1	relopabi 4730	1 ⊢ Rel {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 103   = wceq 1343  ∃wex 1480  ⟨cop 3579  Rel wrel 4609  {coprab 5843

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-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  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-opab 4044  df-xp 4610  df-rel 4611  df-oprab 5846

This theorem is referenced by:  oprabss  5928