Theorem resid 5038

Description: Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)

Assertion

Ref	Expression
resid	⊢ (Rel 𝐴 → (𝐴 ↾ V) = 𝐴)

Proof of Theorem resid

Step	Hyp	Ref	Expression
1		ssv 3226	. 2 ⊢ dom 𝐴 ⊆ V
2		relssres 5019	. 2 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ V) → (𝐴 ↾ V) = 𝐴)
3	1, 2	mpan2 425	1 ⊢ (Rel 𝐴 → (𝐴 ↾ V) = 𝐴)

Syntax hints:   → wi 4   = wceq 1375  Vcvv 2779   ⊆ wss 3177  dom cdm 4696   ↾ cres 4698  Rel wrel 4701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-br 4063  df-opab 4125  df-xp 4702  df-rel 4703  df-dm 4706  df-res 4708

This theorem is referenced by: (None)