Theorem cores 4767

Description: Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
cores	⊢ (ran B ⊆ 𝐶 → ((A ↾ 𝐶) ∘ B) = (A ∘ B))

Proof of Theorem cores

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2554	. . . . . . 7 ⊢ z ∈ V
2		vex 2554	. . . . . . 7 ⊢ y ∈ V
3	1, 2	brelrn 4510	. . . . . 6 ⊢ (zBy → y ∈ ran B)
4		ssel 2933	. . . . . 6 ⊢ (ran B ⊆ 𝐶 → (y ∈ ran B → y ∈ 𝐶))
5		vex 2554	. . . . . . . 8 ⊢ x ∈ V
6	5	brres 4561	. . . . . . 7 ⊢ (y(A ↾ 𝐶)x ↔ (yAx ∧ y ∈ 𝐶))
7	6	rbaib 829	. . . . . 6 ⊢ (y ∈ 𝐶 → (y(A ↾ 𝐶)x ↔ yAx))
8	3, 4, 7	syl56 30	. . . . 5 ⊢ (ran B ⊆ 𝐶 → (zBy → (y(A ↾ 𝐶)x ↔ yAx)))
9	8	pm5.32d 423	. . . 4 ⊢ (ran B ⊆ 𝐶 → ((zBy ∧ y(A ↾ 𝐶)x) ↔ (zBy ∧ yAx)))
10	9	exbidv 1703	. . 3 ⊢ (ran B ⊆ 𝐶 → (∃y(zBy ∧ y(A ↾ 𝐶)x) ↔ ∃y(zBy ∧ yAx)))
11	10	opabbidv 3814	. 2 ⊢ (ran B ⊆ 𝐶 → {⟨z, x⟩ ∣ ∃y(zBy ∧ y(A ↾ 𝐶)x)} = {⟨z, x⟩ ∣ ∃y(zBy ∧ yAx)})
12		df-co 4297	. 2 ⊢ ((A ↾ 𝐶) ∘ B) = {⟨z, x⟩ ∣ ∃y(zBy ∧ y(A ↾ 𝐶)x)}
13		df-co 4297	. 2 ⊢ (A ∘ B) = {⟨z, x⟩ ∣ ∃y(zBy ∧ yAx)}
14	11, 12, 13	3eqtr4g 2094	1 ⊢ (ran B ⊆ 𝐶 → ((A ↾ 𝐶) ∘ B) = (A ∘ B))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390   ⊆ wss 2911   class class class wbr 3755  {copab 3808  ran crn 4289   ↾ cres 4290   ∘ ccom 4292

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300

This theorem is referenced by:  cocnvcnv1  4774  cores2  4776  relcoi2  4791  fco2  5000  fcoi2  5014