Theorem coeq1 4875

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

Assertion

Ref	Expression
coeq1	⊢ (A = B → (A ∘ C) = (B ∘ C))

Proof of Theorem coeq1

Step	Hyp	Ref	Expression
1		coss1 4873	. . 3 ⊢ (A ⊆ B → (A ∘ C) ⊆ (B ∘ C))
2		coss1 4873	. . 3 ⊢ (B ⊆ A → (B ∘ C) ⊆ (A ∘ C))
3	1, 2	anim12i 549	. 2 ⊢ ((A ⊆ B ∧ B ⊆ A) → ((A ∘ C) ⊆ (B ∘ C) ∧ (B ∘ C) ⊆ (A ∘ C)))
4		eqss 3288	. 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
5		eqss 3288	. 2 ⊢ ((A ∘ C) = (B ∘ C) ↔ ((A ∘ C) ⊆ (B ∘ C) ∧ (B ∘ C) ⊆ (A ∘ C)))
6	3, 4, 5	3imtr4i 257	1 ⊢ (A = B → (A ∘ C) = (B ∘ C))

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:  coeq1i  4877  coeq1d  4879  composevalg  5818  pprodeq1  5835  pprodeq2  5836  enmap2lem3  6066  enmap2lem5  6068