Theorem reseq2 5008

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 4739	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V))
2	1	ineq2d 3408	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V)))
3		df-res 4737	. 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
4		df-res 4737	. 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
5	2, 3, 4	3eqtr4g 2289	1 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1397  Vcvv 2802   ∩ cin 3199   × cxp 4723   ↾ cres 4727

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-opab 4151  df-xp 4731  df-res 4737

This theorem is referenced by:  reseq2i  5010  reseq2d  5013  resabs1  5042  resima2  5047  imaeq2  5072  resdisj  5165  relcoi1  5268  fressnfv  5840  tfrlem1  6473  tfrlem9  6484  tfr0dm  6487  tfrlemisucaccv  6490  tfrlemiubacc  6495  tfr1onlemsucaccv  6506  tfr1onlemubacc  6511  tfr1onlemaccex  6513  tfrcllemsucaccv  6519  tfrcllembxssdm  6521  tfrcllemubacc  6524  tfrcllemaccex  6526  tfrcllemres  6527  tfrcldm  6528  fnfi  7134  lmbr2  14937  lmff  14972  dvmptid  15439