Theorem coeq2 4876

Description: Equality theorem for composition of two classes. (Contributed by set.mm contributors, 3-Jan-1997.)

Assertion

Ref	Expression
coeq2	⊢ (A = B → (C ∘ A) = (C ∘ B))

Proof of Theorem coeq2

Step	Hyp	Ref	Expression
1		coss2 4874	. . 3 ⊢ (A ⊆ B → (C ∘ A) ⊆ (C ∘ B))
2		coss2 4874	. . 3 ⊢ (B ⊆ A → (C ∘ B) ⊆ (C ∘ A))
3	1, 2	anim12i 549	. 2 ⊢ ((A ⊆ B ∧ B ⊆ A) → ((C ∘ A) ⊆ (C ∘ B) ∧ (C ∘ B) ⊆ (C ∘ A)))
4		eqss 3288	. 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
5		eqss 3288	. 2 ⊢ ((C ∘ A) = (C ∘ B) ↔ ((C ∘ A) ⊆ (C ∘ B) ∧ (C ∘ B) ⊆ (C ∘ A)))
6	3, 4, 5	3imtr4i 257	1 ⊢ (A = B → (C ∘ A) = (C ∘ B))

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ⊆ wss 3258   ∘ ccom 4722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-opab 4624  df-br 4641  df-co 4727

This theorem is referenced by:  coeq2i  4878  coeq2d  4880  txpeq1  5780  txpeq2  5781  composevalg  5818  enmap1lem3  6072  enmap1lem5  6074