Theorem fcoi1 2765

Description: Composition of a mapping and restricted identity.

Assertion

Ref	Expression
fcoi1	⊢ (F:A–→B → (F ∘ (I ↾ A)) = F)

Proof of Theorem fcoi1

Step	Hyp	Ref	Expression
1		ffn 2752	. 2 ⊢ (F:A–→B → F Fn A)
2		visset 1350	. . . . . . . 8 ⊢ x ∈ V
3		visset 1350	. . . . . . . 8 ⊢ y ∈ V
4	2, 3	fnop 2727	. . . . . . 7 ⊢ ((F Fn A ∧ ⟨x, y⟩ ∈ F) → x ∈ A)
5	4	exp 291	. . . . . 6 ⊢ (F Fn A → (⟨x, y⟩ ∈ F → x ∈ A))
6		pm4.71r 482	. . . . . 6 ⊢ ((⟨x, y⟩ ∈ F → x ∈ A) ↔ (⟨x, y⟩ ∈ F ↔ (x ∈ A ∧ ⟨x, y⟩ ∈ F)))
7	5, 6	sylib 173	. . . . 5 ⊢ (F Fn A → (⟨x, y⟩ ∈ F ↔ (x ∈ A ∧ ⟨x, y⟩ ∈ F)))
8		opelcog 2511	. . . . . . 7 ⊢ ((x ∈ V ∧ y ∈ V) → (⟨x, y⟩ ∈ (F ∘ (I ↾ A)) ↔ ∃z(⟨x, z⟩ ∈ (I ↾ A) ∧ ⟨z, y⟩ ∈ F)))
9	2, 3, 8	mp2an 520	. . . . . 6 ⊢ (⟨x, y⟩ ∈ (F ∘ (I ↾ A)) ↔ ∃z(⟨x, z⟩ ∈ (I ↾ A) ∧ ⟨z, y⟩ ∈ F))
10		visset 1350	. . . . . . . . . . 11 ⊢ z ∈ V
11	10	opelres 2579	. . . . . . . . . 10 ⊢ (⟨x, z⟩ ∈ (I ↾ A) ↔ (⟨x, z⟩ ∈ I ∧ x ∈ A))
12	2, 10	ideq 2127	. . . . . . . . . . . 12 ⊢ (xIz ↔ x = z)
13		df-br 2063	. . . . . . . . . . . 12 ⊢ (xIz ↔ ⟨x, z⟩ ∈ I)
14		cleqcom 1103	. . . . . . . . . . . 12 ⊢ (x = z ↔ z = x)
15	12, 13, 14	3bitr3 156	. . . . . . . . . . 11 ⊢ (⟨x, z⟩ ∈ I ↔ z = x)
16	15	anbi1i 368	. . . . . . . . . 10 ⊢ ((⟨x, z⟩ ∈ I ∧ x ∈ A) ↔ (z = x ∧ x ∈ A))
17	11, 16	bitr 151	. . . . . . . . 9 ⊢ (⟨x, z⟩ ∈ (I ↾ A) ↔ (z = x ∧ x ∈ A))
18	17	anbi1i 368	. . . . . . . 8 ⊢ ((⟨x, z⟩ ∈ (I ↾ A) ∧ ⟨z, y⟩ ∈ F) ↔ ((z = x ∧ x ∈ A) ∧ ⟨z, y⟩ ∈ F))
19		anass 336	. . . . . . . 8 ⊢ (((z = x ∧ x ∈ A) ∧ ⟨z, y⟩ ∈ F) ↔ (z = x ∧ (x ∈ A ∧ ⟨z, y⟩ ∈ F)))
20	18, 19	bitr 151	. . . . . . 7 ⊢ ((⟨x, z⟩ ∈ (I ↾ A) ∧ ⟨z, y⟩ ∈ F) ↔ (z = x ∧ (x ∈ A ∧ ⟨z, y⟩ ∈ F)))
21	20	biex 733	. . . . . 6 ⊢ (∃z(⟨x, z⟩ ∈ (I ↾ A) ∧ ⟨z, y⟩ ∈ F) ↔ ∃z(z = x ∧ (x ∈ A ∧ ⟨z, y⟩ ∈ F)))
22		opeq1 1876	. . . . . . . . 9 ⊢ (z = x → ⟨z, y⟩ = ⟨x, y⟩)
23	22	eleq1d 1155	. . . . . . . 8 ⊢ (z = x → (⟨z, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ F))
24	23	anbi2d 468	. . . . . . 7 ⊢ (z = x → ((x ∈ A ∧ ⟨z, y⟩ ∈ F) ↔ (x ∈ A ∧ ⟨x, y⟩ ∈ F)))
25	2, 24	ceqsexv 1371	. . . . . 6 ⊢ (∃z(z = x ∧ (x ∈ A ∧ ⟨z, y⟩ ∈ F)) ↔ (x ∈ A ∧ ⟨x, y⟩ ∈ F))
26	9, 21, 25	3bitr 155	. . . . 5 ⊢ (⟨x, y⟩ ∈ (F ∘ (I ↾ A)) ↔ (x ∈ A ∧ ⟨x, y⟩ ∈ F))
27	7, 26	syl6rbbr 417	. . . 4 ⊢ (F Fn A → (⟨x, y⟩ ∈ (F ∘ (I ↾ A)) ↔ ⟨x, y⟩ ∈ F))
28	27	19.21aivv 944	. . 3 ⊢ (F Fn A → ∀x∀y(⟨x, y⟩ ∈ (F ∘ (I ↾ A)) ↔ ⟨x, y⟩ ∈ F))
29		fnrel 2722	. . . 4 ⊢ (F Fn A → Rel F)
30		relco 2658	. . . . 5 ⊢ Rel (F ∘ (I ↾ A))
31		cleqrel 2483	. . . . 5 ⊢ ((Rel (F ∘ (I ↾ A)) ∧ Rel F) → ((F ∘ (I ↾ A)) = F ↔ ∀x∀y(⟨x, y⟩ ∈ (F ∘ (I ↾ A)) ↔ ⟨x, y⟩ ∈ F)))
32	30, 31	mpan 518	. . . 4 ⊢ (Rel F → ((F ∘ (I ↾ A)) = F ↔ ∀x∀y(⟨x, y⟩ ∈ (F ∘ (I ↾ A)) ↔ ⟨x, y⟩ ∈ F)))
33	29, 32	syl 12	. . 3 ⊢ (F Fn A → ((F ∘ (I ↾ A)) = F ↔ ∀x∀y(⟨x, y⟩ ∈ (F ∘ (I ↾ A)) ↔ ⟨x, y⟩ ∈ F)))
34	28, 33	mpbird 171	. 2 ⊢ (F Fn A → (F ∘ (I ↾ A)) = F)
35	1, 34	syl 12	1 ⊢ (F:A–→B → (F ∘ (I ↾ A)) = F)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   class class class wbr 2054  Icid 2057   ↾ cres 2412   ∘ ccom 2414  Rel wrel 2415   Fn wfn 2417  –→wf 2418

This theorem is referenced by:  mapenlem1 3384  mapenlem2 3385

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-id 2125  df-xp 2424  df-rel 2425  df-co 2427  df-dm 2428  df-res 2430  df-fun 2432  df-fn 2433  df-f 2434

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