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Theorem coeq2 4437
Description: Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
Assertion
Ref Expression
coeq2 (A = B → (𝐶A) = (𝐶B))

Proof of Theorem coeq2
StepHypRef Expression
1 coss2 4435 . . 3 (AB → (𝐶A) ⊆ (𝐶B))
2 coss2 4435 . . 3 (BA → (𝐶B) ⊆ (𝐶A))
31, 2anim12i 321 . 2 ((AB BA) → ((𝐶A) ⊆ (𝐶B) (𝐶B) ⊆ (𝐶A)))
4 eqss 2954 . 2 (A = B ↔ (AB BA))
5 eqss 2954 . 2 ((𝐶A) = (𝐶B) ↔ ((𝐶A) ⊆ (𝐶B) (𝐶B) ⊆ (𝐶A)))
63, 4, 53imtr4i 190 1 (A = B → (𝐶A) = (𝐶B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wss 2911  ccom 4292
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925  df-br 3756  df-opab 3810  df-co 4297
This theorem is referenced by:  coeq2i  4439  coeq2d  4441  coi2  4780  relcnvtr  4783  relcoi1  4792  f1eqcocnv  5374  ereq1  6049
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