Theorem reseq2 4963

Description: Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)

Assertion

Ref	Expression
reseq2	⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵))

Proof of Theorem reseq2

Step	Hyp	Ref	Expression
1		xpeq1 4697	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V))
2	1	ineq2d 3378	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V)))
3		df-res 4695	. 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
4		df-res 4695	. 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
5	2, 3, 4	3eqtr4g 2264	1 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1373  Vcvv 2773   ∩ cin 3169   × cxp 4681   ↾ cres 4685

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3176  df-opab 4114  df-xp 4689  df-res 4695

This theorem is referenced by:  reseq2i  4965  reseq2d  4968  resabs1  4997  resima2  5002  imaeq2  5027  resdisj  5120  relcoi1  5223  fressnfv  5784  tfrlem1  6407  tfrlem9  6418  tfr0dm  6421  tfrlemisucaccv  6424  tfrlemiubacc  6429  tfr1onlemsucaccv  6440  tfr1onlemubacc  6445  tfr1onlemaccex  6447  tfrcllemsucaccv  6453  tfrcllembxssdm  6455  tfrcllemubacc  6458  tfrcllemaccex  6460  tfrcllemres  6461  tfrcldm  6462  fnfi  7053  lmbr2  14761  lmff  14796  dvmptid  15263