Theorem coeq2 3282

Description: Equality theorem for composition of two classes.

Assertion

Ref	Expression
coeq2	|- (A = B -> (C o. A) = (C o. B))

Proof of Theorem coeq2

Step	Hyp	Ref	Expression
1		breq 2621	. . . . 5 |- (A = B -> (xAz <-> xBz))
2	1	anbi1d 617	. . . 4 |- (A = B -> ((xAz /\ zCy) <-> (xBz /\ zCy)))
3	2	exbidv 1279	. . 3 |- (A = B -> (E.z(xAz /\ zCy) <-> E.z(xBz /\ zCy)))
4	3	opabbidv 2670	. 2 |- (A = B -> {<.x, y>. | E.z(xAz /\ zCy)} = {<.x, y>. | E.z(xBz /\ zCy)})
5		df-co 3187	. 2 |- (C o. A) = {<.x, y>. | E.z(xAz /\ zCy)}
6		df-co 3187	. 2 |- (C o. B) = {<.x, y>. | E.z(xBz /\ zCy)}
7	4, 5, 6	3eqtr4g 1531	1 |- (A = B -> (C o. A) = (C o. B))

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:  coeq2i 3284  coeq2d 3286  coi2 3511  ereq 4267  mapenlem1 4489  mapenlem2 4490  isps 8645  hocsubdirt 9711  hoddit 9915  lnopco0 9929  hmopidmcht 10081  hmopidmpjt 10082  pjsdi2 10085  pjidmcot 10109  pjhmopidm 10110  dfpjopt 10111  pjin2 10121  pjclem1 10123  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

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