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Theorem coeq2 4421
 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 4419 . . 3 (AB → (𝐶A) ⊆ (𝐶B))
2 coss2 4419 . . 3 (BA → (𝐶B) ⊆ (𝐶A))
31, 2anim12i 321 . 2 ((AB BA) → ((𝐶A) ⊆ (𝐶B) (𝐶B) ⊆ (𝐶A)))
4 eqss 2937 . 2 (A = B ↔ (AB BA))
5 eqss 2937 . 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 1228   ⊆ wss 2894   ∘ ccom 4276 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-in 2901  df-ss 2908  df-br 3739  df-opab 3793  df-co 4281 This theorem is referenced by:  coeq2i  4423  coeq2d  4425  coi2  4764  relcnvtr  4767  relcoi1  4776  f1eqcocnv  5356  ereq1  6024
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