Theorem reseq2 4873

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 4612	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V))
2	1	ineq2d 3318	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V)))
3		df-res 4610	. 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
4		df-res 4610	. 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
5	2, 3, 4	3eqtr4g 2222	1 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1342  Vcvv 2721   ∩ cin 3110   × cxp 4596   ↾ cres 4600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-in 3117  df-opab 4038  df-xp 4604  df-res 4610

This theorem is referenced by:  reseq2i  4875  reseq2d  4878  resabs1  4907  resima2  4912  imaeq2  4936  resdisj  5026  relcoi1  5129  fressnfv  5666  tfrlem1  6267  tfrlem9  6278  tfr0dm  6281  tfrlemisucaccv  6284  tfrlemiubacc  6289  tfr1onlemsucaccv  6300  tfr1onlemubacc  6305  tfr1onlemaccex  6307  tfrcllemsucaccv  6313  tfrcllembxssdm  6315  tfrcllemubacc  6318  tfrcllemaccex  6320  tfrcllemres  6321  tfrcldm  6322  fnfi  6893  lmbr2  12755  lmff  12790