Theorem reseq2 4942

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 4678	. . 3 |- ( A = B -> ( A X. _V ) = ( B X. _V ) )
2	1	ineq2d 3365	. 2 |- ( A = B -> ( C i^i ( A X. _V ) ) = ( C i^i ( B X. _V ) ) )
3		df-res 4676	. 2 |- ( C |` A ) = ( C i^i ( A X. _V ) )
4		df-res 4676	. 2 |- ( C |` B ) = ( C i^i ( B X. _V ) )
5	2, 3, 4	3eqtr4g 2254	1 |- ( A = B -> ( C |` A ) = ( C |` B ) )

Syntax hints:   -> wi 4   = wceq 1364   _Vcvv 2763   i^i cin 3156   X. cxp 4662   |` cres 4666

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 4096  df-xp 4670  df-res 4676

This theorem is referenced by:  reseq2i  4944  reseq2d  4947  resabs1  4976  resima2  4981  imaeq2  5006  resdisj  5099  relcoi1  5202  fressnfv  5752  tfrlem1  6375  tfrlem9  6386  tfr0dm  6389  tfrlemisucaccv  6392  tfrlemiubacc  6397  tfr1onlemsucaccv  6408  tfr1onlemubacc  6413  tfr1onlemaccex  6415  tfrcllemsucaccv  6421  tfrcllembxssdm  6423  tfrcllemubacc  6426  tfrcllemaccex  6428  tfrcllemres  6429  tfrcldm  6430  fnfi  7011  lmbr2  14534  lmff  14569  dvmptid  15036