Theorem reseq2 5038

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 4768	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V))
2	1	ineq2d 3426	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V)))
3		df-res 4766	. 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
4		df-res 4766	. 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
5	2, 3, 4	3eqtr4g 2292	1 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1398  Vcvv 2815   ∩ cin 3213   × cxp 4752   ↾ cres 4756

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3220  df-opab 4177  df-xp 4760  df-res 4766

This theorem is referenced by:  reseq2i  5040  reseq2d  5043  resabs1  5072  resima2  5077  imaeq2  5102  resdisj  5196  relcoi1  5299  fressnfv  5876  tfrlem1  6552  tfrlem9  6563  tfr0dm  6566  tfrlemisucaccv  6569  tfrlemiubacc  6574  tfr1onlemsucaccv  6585  tfr1onlemubacc  6590  tfr1onlemaccex  6592  tfrcllemsucaccv  6598  tfrcllembxssdm  6600  tfrcllemubacc  6603  tfrcllemaccex  6605  tfrcllemres  6606  tfrcldm  6607  fnfi  7216  gfsumcl  14110  lmbr2  15205  lmff  15240  dvmptid  15707