Theorem reseq2 5060

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 4790	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V))
2	1	ineq2d 3432	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V)))
3		df-res 4788	. 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
4		df-res 4788	. 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
5	2, 3, 4	3eqtr4g 2296	1 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1402  Vcvv 2821   ∩ cin 3219   × cxp 4774   ↾ cres 4778

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-opab 4194  df-xp 4782  df-res 4788

This theorem is used by:  reseq2i  5062  reseq2d  5065  resabs1  5094  resima2  5099  imaeq2  5125  resdisj  5219  relcoi1  5322  fressnfv  5905  tfrlem1  6583  tfrlem9  6594  tfr0dm  6597  tfrlemisucaccv  6600  tfrlemiubacc  6605  tfr1onlemsucaccv  6616  tfr1onlemubacc  6621  tfr1onlemaccex  6623  tfrcllemsucaccv  6629  tfrcllembxssdm  6631  tfrcllemubacc  6634  tfrcllemaccex  6636  tfrcllemres  6637  tfrcldm  6638  fnfi  7254  gsumclfi  14249  gsummptfidmadd  14251  gsumsubmclfi  14253  gsumresfi  14263  gsum2dfilem2  14272  lmbr2  15440  lmff  15475  dvmptid  15942