Theorem reldom 7027

Description: Dominance is a relation. (Contributed by NM, 28-Mar-1998.)

Assertion

Ref	Expression
reldom	⊢ Rel ≼

Proof of Theorem reldom

Dummy variables 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dom 7024	. 2 ⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
2	1	relopabi 4905	1 ⊢ Rel ≼

This proof depends on syntax axioms:  ∃wex 1545  Rel wrel 4779  –1-1→wf1 5374   ≼ cdom 7021

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-opab 4193  df-xp 4780  df-rel 4781  df-dom 7024

This theorem is used by:  brdomg  7032  brdomi  7033  ctex  7037  domssr  7064  domtr  7072  xpdom2  7129  xpdom1g  7131  mapdom1g  7147  isbth  7284  djudom  7433  difinfsn  7440  hashinfom  11219