Theorem coeq2 2503

Description: Equality theorem for composition of two classes.

Assertion

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

Proof of Theorem coeq2

Step	Hyp	Ref	Expression
1		breq 2064	. . . . 5 ⊢ (A = B → (xAz ↔ xBz))
2	1	anbi1d 469	. . . 4 ⊢ (A = B → ((xAz ∧ zCy) ↔ (xBz ∧ zCy)))
3	2	biexdv 936	. . 3 ⊢ (A = B → (∃z(xAz ∧ zCy) ↔ ∃z(xBz ∧ zCy)))
4	3	biopabdv 2102	. 2 ⊢ (A = B → {⟨x, y⟩∣∃z(xAz ∧ zCy)} = {⟨x, y⟩∣∃z(xBz ∧ zCy)})
5		df-co 2427	. 2 ⊢ (C ∘ A) = {⟨x, y⟩∣∃z(xAz ∧ zCy)}
6		df-co 2427	. 2 ⊢ (C ∘ B) = {⟨x, y⟩∣∃z(xBz ∧ zCy)}
7	4, 5, 6	3eqtr4g 1147	1 ⊢ (A = B → (C ∘ A) = (C ∘ B))

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678   = wceq 1091   class class class wbr 2054  {copab 2055   ∘ ccom 2414

This theorem is referenced by:  coeq2i 2505  coeq2d 2507  coi2 2666  ereq 3206  mapenlem1 3384  mapenlem2 3385  pjsdi2 5627  pjin2 5647  pjclem1 5649

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-br 2063  df-opab 2098  df-co 2427

metamath.org