Theorem reseq2 4941

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 4677	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V))
2	1	ineq2d 3364	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V)))
3		df-res 4675	. 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
4		df-res 4675	. 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
5	2, 3, 4	3eqtr4g 2254	1 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1364  Vcvv 2763   ∩ cin 3156   × cxp 4661   ↾ cres 4665

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-opab 4095  df-xp 4669  df-res 4675

This theorem is referenced by:  reseq2i  4943  reseq2d  4946  resabs1  4975  resima2  4980  imaeq2  5005  resdisj  5098  relcoi1  5201  fressnfv  5749  tfrlem1  6366  tfrlem9  6377  tfr0dm  6380  tfrlemisucaccv  6383  tfrlemiubacc  6388  tfr1onlemsucaccv  6399  tfr1onlemubacc  6404  tfr1onlemaccex  6406  tfrcllemsucaccv  6412  tfrcllembxssdm  6414  tfrcllemubacc  6417  tfrcllemaccex  6419  tfrcllemres  6420  tfrcldm  6421  fnfi  7002  lmbr2  14450  lmff  14485  dvmptid  14952