Theorem resid 5100

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 3264	. 2 ⊢ dom 𝐴 ⊆ V
2		relssres 5081	. 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 2815   ⊆ wss 3214  dom cdm 4754   ↾ cres 4756  Rel wrel 4759

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 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761  df-dm 4764  df-res 4766

This theorem is referenced by: (None)