Theorem coeq2 3271

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 2611	. . . . 5 |- (A = B -> (xAz <-> xBz))
2	1	anbi1d 615	. . . 4 |- (A = B -> ((xAz /\ zCy) <-> (xBz /\ zCy)))
3	2	exbidv 1274	. . 3 |- (A = B -> (E.z(xAz /\ zCy) <-> E.z(xBz /\ zCy)))
4	3	opabbidv 2660	. 2 |- (A = B -> {<.x, y>. | E.z(xAz /\ zCy)} = {<.x, y>. | E.z(xBz /\ zCy)})
5		df-co 3177	. 2 |- (C o. A) = {<.x, y>. | E.z(xAz /\ zCy)}
6		df-co 3177	. 2 |- (C o. B) = {<.x, y>. | E.z(xBz /\ zCy)}
7	4, 5, 6	3eqtr4g 1523	1 |- (A = B -> (C o. A) = (C o. B))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953  E.wex 977   class class class wbr 2609  {copab 2656   o. ccom 3164

This theorem is referenced by:  coeq2i 3273  coeq2d 3275  coi2 3497  ereq 4251  mapenlem1 4469  mapenlem2 4470  isps 8571  hocsubdirt 9628  hoddit 9830  lnopco0 9844  hmopidmcht 9992  hmopidmpjt 9993  pjsdi2 9996  pjidmcot 10019  pjhmopidm 10020  dfpjopt 10021  pjin2 10031  pjclem1 10033  symgoprval 10309  hmeogrp 10425

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-br 2610  df-opab 2657  df-co 3177

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