Theorem resid 5094

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 3259	. 2 ⊢ dom 𝐴 ⊆ V
2		relssres 5075	. 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 1398  Vcvv 2812   ⊆ wss 3210  dom cdm 4748   ↾ cres 4750  Rel wrel 4753

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171  df-xp 4754  df-rel 4755  df-dm 4758  df-res 4760

This theorem is referenced by: (None)