Theorem reseq2 4607

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 4359	. . 3 |- ( A = B -> ( A X. _V ) = ( B X. _V ) )
2	1	ineq2d 3138	. 2 |- ( A = B -> ( C i^i ( A X. _V ) ) = ( C i^i ( B X. _V ) ) )
3		df-res 4357	. 2 |- ( C |` A ) = ( C i^i ( A X. _V ) )
4		df-res 4357	. 2 |- ( C |` B ) = ( C i^i ( B X. _V ) )
5	2, 3, 4	3eqtr4g 2097	1 |- ( A = B -> ( C |` A ) = ( C |` B ) )

Syntax hints:   -> wi 4   = wceq 1243   _Vcvv 2557   i^i cin 2916   X. cxp 4343   |` cres 4347

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-opab 3819  df-xp 4351  df-res 4357

This theorem is referenced by:  reseq2i  4609  reseq2d  4612  resabs1  4640  resima2  4644  imaeq2  4664  resdisj  4751  relcoi1  4849  fressnfv  5350  tfrlem1  5923  tfrlem9  5935  tfr0  5937  tfrlemisucaccv  5939  tfrlemiubacc  5944