Theorem cores 2659

Description: The first member of a composition may be restricted down to the range of the second without affecting the result.

Assertion

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

Proof of Theorem cores

Step	Hyp	Ref	Expression
1		pm3.26 256	. . . . . . . . 9 ⊢ ((xBz ∧ z ∈ C) → xBz)
2	1	anim1i 269	. . . . . . . 8 ⊢ (((xBz ∧ z ∈ C) ∧ zAy) → (xBz ∧ zAy))
3	2	a1i 7	. . . . . . 7 ⊢ (ran B ⊆ C → (((xBz ∧ z ∈ C) ∧ zAy) → (xBz ∧ zAy)))
4		ssel 1502	. . . . . . . . . 10 ⊢ (ran B ⊆ C → (z ∈ ran B → z ∈ C))
5		visset 1350	. . . . . . . . . . 11 ⊢ x ∈ V
6		visset 1350	. . . . . . . . . . 11 ⊢ z ∈ V
7	5, 6	brelrn 2559	. . . . . . . . . 10 ⊢ (xBz → z ∈ ran B)
8	4, 7	syl5 22	. . . . . . . . 9 ⊢ (ran B ⊆ C → (xBz → z ∈ C))
9	8	ancld 246	. . . . . . . 8 ⊢ (ran B ⊆ C → (xBz → (xBz ∧ z ∈ C)))
10	9	anim1d 432	. . . . . . 7 ⊢ (ran B ⊆ C → ((xBz ∧ zAy) → ((xBz ∧ z ∈ C) ∧ zAy)))
11	3, 10	impbid 397	. . . . . 6 ⊢ (ran B ⊆ C → (((xBz ∧ z ∈ C) ∧ zAy) ↔ (xBz ∧ zAy)))
12		visset 1350	. . . . . . . . . . 11 ⊢ y ∈ V
13	12	opelres 2579	. . . . . . . . . 10 ⊢ (⟨z, y⟩ ∈ (A ↾ C) ↔ (⟨z, y⟩ ∈ A ∧ z ∈ C))
14		df-br 2063	. . . . . . . . . 10 ⊢ (z(A ↾ C)y ↔ ⟨z, y⟩ ∈ (A ↾ C))
15		df-br 2063	. . . . . . . . . . 11 ⊢ (zAy ↔ ⟨z, y⟩ ∈ A)
16	15	anbi1i 368	. . . . . . . . . 10 ⊢ ((zAy ∧ z ∈ C) ↔ (⟨z, y⟩ ∈ A ∧ z ∈ C))
17	13, 14, 16	3bitr4 158	. . . . . . . . 9 ⊢ (z(A ↾ C)y ↔ (zAy ∧ z ∈ C))
18		ancom 333	. . . . . . . . 9 ⊢ ((zAy ∧ z ∈ C) ↔ (z ∈ C ∧ zAy))
19	17, 18	bitr 151	. . . . . . . 8 ⊢ (z(A ↾ C)y ↔ (z ∈ C ∧ zAy))
20	19	anbi2i 367	. . . . . . 7 ⊢ ((xBz ∧ z(A ↾ C)y) ↔ (xBz ∧ (z ∈ C ∧ zAy)))
21		anass 336	. . . . . . 7 ⊢ (((xBz ∧ z ∈ C) ∧ zAy) ↔ (xBz ∧ (z ∈ C ∧ zAy)))
22	20, 21	bitr4 154	. . . . . 6 ⊢ ((xBz ∧ z(A ↾ C)y) ↔ ((xBz ∧ z ∈ C) ∧ zAy))
23	11, 22	syl5bb 410	. . . . 5 ⊢ (ran B ⊆ C → ((xBz ∧ z(A ↾ C)y) ↔ (xBz ∧ zAy)))
24	23	biexdv 936	. . . 4 ⊢ (ran B ⊆ C → (∃z(xBz ∧ z(A ↾ C)y) ↔ ∃z(xBz ∧ zAy)))
25	5, 12	opelco 2509	. . . 4 ⊢ (⟨x, y⟩ ∈ ((A ↾ C) ∘ B) ↔ ∃z(xBz ∧ z(A ↾ C)y))
26	5, 12	opelco 2509	. . . 4 ⊢ (⟨x, y⟩ ∈ (A ∘ B) ↔ ∃z(xBz ∧ zAy))
27	24, 25, 26	3bitr4g 428	. . 3 ⊢ (ran B ⊆ C → (⟨x, y⟩ ∈ ((A ↾ C) ∘ B) ↔ ⟨x, y⟩ ∈ (A ∘ B)))
28	27	19.21aivv 944	. 2 ⊢ (ran B ⊆ C → ∀x∀y(⟨x, y⟩ ∈ ((A ↾ C) ∘ B) ↔ ⟨x, y⟩ ∈ (A ∘ B)))
29		relco 2658	. . 3 ⊢ Rel ((A ↾ C) ∘ B)
30		relco 2658	. . 3 ⊢ Rel (A ∘ B)
31		cleqrel 2483	. . 3 ⊢ ((Rel ((A ↾ C) ∘ B) ∧ Rel (A ∘ B)) → (((A ↾ C) ∘ B) = (A ∘ B) ↔ ∀x∀y(⟨x, y⟩ ∈ ((A ↾ C) ∘ B) ↔ ⟨x, y⟩ ∈ (A ∘ B))))
32	29, 30, 31	mp2an 520	. 2 ⊢ (((A ↾ C) ∘ B) = (A ∘ B) ↔ ∀x∀y(⟨x, y⟩ ∈ ((A ↾ C) ∘ B) ↔ ⟨x, y⟩ ∈ (A ∘ B)))
33	28, 32	sylibr 175	1 ⊢ (ran B ⊆ C → ((A ↾ C) ∘ B) = (A ∘ B))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  ⟨cop 1810   class class class wbr 2054  ran crn 2411   ↾ cres 2412   ∘ ccom 2414  Rel wrel 2415

This theorem is referenced by:  ruclem17 4901

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430

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