Theorem reseq2 4973

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 4707	. . 3 |- ( A = B -> ( A X. _V ) = ( B X. _V ) )
2	1	ineq2d 3382	. 2 |- ( A = B -> ( C i^i ( A X. _V ) ) = ( C i^i ( B X. _V ) ) )
3		df-res 4705	. 2 |- ( C |` A ) = ( C i^i ( A X. _V ) )
4		df-res 4705	. 2 |- ( C |` B ) = ( C i^i ( B X. _V ) )
5	2, 3, 4	3eqtr4g 2265	1 |- ( A = B -> ( C |` A ) = ( C |` B ) )

Syntax hints:   -> wi 4   = wceq 1373   _Vcvv 2776   i^i cin 3173   X. cxp 4691   |` cres 4695

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-opab 4122  df-xp 4699  df-res 4705

This theorem is referenced by:  reseq2i  4975  reseq2d  4978  resabs1  5007  resima2  5012  imaeq2  5037  resdisj  5130  relcoi1  5233  fressnfv  5794  tfrlem1  6417  tfrlem9  6428  tfr0dm  6431  tfrlemisucaccv  6434  tfrlemiubacc  6439  tfr1onlemsucaccv  6450  tfr1onlemubacc  6455  tfr1onlemaccex  6457  tfrcllemsucaccv  6463  tfrcllembxssdm  6465  tfrcllemubacc  6468  tfrcllemaccex  6470  tfrcllemres  6471  tfrcldm  6472  fnfi  7064  lmbr2  14801  lmff  14836  dvmptid  15303