Theorem coeq1 3281

Description: Equality theorem for composition of two classes.

Assertion

Ref	Expression
coeq1	|- (A = B -> (A o. C) = (B o. C))

Proof of Theorem coeq1

Step	Hyp	Ref	Expression
1		breq 2621	. . . . 5 |- (A = B -> (zAy <-> zBy))
2	1	anbi2d 616	. . . 4 |- (A = B -> ((xCz /\ zAy) <-> (xCz /\ zBy)))
3	2	exbidv 1279	. . 3 |- (A = B -> (E.z(xCz /\ zAy) <-> E.z(xCz /\ zBy)))
4	3	opabbidv 2670	. 2 |- (A = B -> {<.x, y>. | E.z(xCz /\ zAy)} = {<.x, y>. | E.z(xCz /\ zBy)})
5		df-co 3187	. 2 |- (A o. C) = {<.x, y>. | E.z(xCz /\ zAy)}
6		df-co 3187	. 2 |- (B o. C) = {<.x, y>. | E.z(xCz /\ zBy)}
7	4, 5, 6	3eqtr4g 1531	1 |- (A = B -> (A o. C) = (B o. C))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956  E.wex 980   class class class wbr 2619  {copab 2666   o. ccom 3174

This theorem is referenced by:  coeq1i 3283  coeq1d 3285  coi2 3511  ereq 4267  isps 8645  hocsubdirt 9711  hoddit 9915  hmopidmcht 10081  hmopidmpjt 10082  pjidmcot 10109  pjhmopidm 10110  dfpjopt 10111  symgoprval 10404  hmeogrp 10538

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-br 2620  df-opab 2667  df-co 3187

Copyright terms: Public domain