Theorem reseq2 4938

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 4674	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V))
2	1	ineq2d 3361	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V)))
3		df-res 4672	. 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
4		df-res 4672	. 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
5	2, 3, 4	3eqtr4g 2251	1 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1364  Vcvv 2760   ∩ cin 3153   × cxp 4658   ↾ cres 4662

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3160  df-opab 4092  df-xp 4666  df-res 4672

This theorem is referenced by:  reseq2i  4940  reseq2d  4943  resabs1  4972  resima2  4977  imaeq2  5002  resdisj  5095  relcoi1  5198  fressnfv  5746  tfrlem1  6363  tfrlem9  6374  tfr0dm  6377  tfrlemisucaccv  6380  tfrlemiubacc  6385  tfr1onlemsucaccv  6396  tfr1onlemubacc  6401  tfr1onlemaccex  6403  tfrcllemsucaccv  6409  tfrcllembxssdm  6411  tfrcllemubacc  6414  tfrcllemaccex  6416  tfrcllemres  6417  tfrcldm  6418  fnfi  6997  lmbr2  14393  lmff  14428  dvmptid  14895