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Theorem coeq1 4416
 Description: Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
Assertion
Ref Expression
coeq1 (A = B → (A𝐶) = (B𝐶))

Proof of Theorem coeq1
StepHypRef Expression
1 coss1 4414 . . 3 (AB → (A𝐶) ⊆ (B𝐶))
2 coss1 4414 . . 3 (BA → (B𝐶) ⊆ (A𝐶))
31, 2anim12i 321 . 2 ((AB BA) → ((A𝐶) ⊆ (B𝐶) (B𝐶) ⊆ (A𝐶)))
4 eqss 2933 . 2 (A = B ↔ (AB BA))
5 eqss 2933 . 2 ((A𝐶) = (B𝐶) ↔ ((A𝐶) ⊆ (B𝐶) (B𝐶) ⊆ (A𝐶)))
63, 4, 53imtr4i 190 1 (A = B → (A𝐶) = (B𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1226   ⊆ wss 2890   ∘ ccom 4272 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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-in 2897  df-ss 2904  df-br 3735  df-opab 3789  df-co 4277 This theorem is referenced by:  coeq1i  4418  coeq1d  4420  coi2  4760  relcnvtr  4763  funcoeqres  5078  ereq1  6020
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