Theorem fcoi1 3651

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 3633	. 2 ⊢ (F:A–→B → F Fn A)
2		fnop 3597	. . . . . . 7 ⊢ ((F Fn A ⋀ ⟨x, y⟩ ∈ F) → x ∈ A)
3	2	ex 373	. . . . . 6 ⊢ (F Fn A → (⟨x, y⟩ ∈ F → x ∈ A))
4	3	pm4.71rd 641	. . . . 5 ⊢ (F Fn A → (⟨x, y⟩ ∈ F ↔ (x ∈ A ⋀ ⟨x, y⟩ ∈ F)))
5		visset 1816	. . . . . . 7 ⊢ x ∈ V
6		visset 1816	. . . . . . 7 ⊢ y ∈ V
7		opelcog 3296	. . . . . . 7 ⊢ ((x ∈ V ⋀ y ∈ V) → (⟨x, y⟩ ∈ (F ∘ (I ↾ A)) ↔ ∃z(⟨x, z⟩ ∈ (I ↾ A) ⋀ ⟨z, y⟩ ∈ F)))
8	5, 6, 7	mp2an 699	. . . . . 6 ⊢ (⟨x, y⟩ ∈ (F ∘ (I ↾ A)) ↔ ∃z(⟨x, z⟩ ∈ (I ↾ A) ⋀ ⟨z, y⟩ ∈ F))
9		visset 1816	. . . . . . . . . . 11 ⊢ z ∈ V
10	9	opelres 3378	. . . . . . . . . 10 ⊢ (⟨x, z⟩ ∈ (I ↾ A) ↔ (⟨x, z⟩ ∈ I ⋀ x ∈ A))
11	9	ideq 3283	. . . . . . . . . . . 12 ⊢ (xIz ↔ x = z)
12		df-br 2625	. . . . . . . . . . . 12 ⊢ (xIz ↔ ⟨x, z⟩ ∈ I)
13		eqcom 1480	. . . . . . . . . . . 12 ⊢ (x = z ↔ z = x)
14	11, 12, 13	3bitr3 181	. . . . . . . . . . 11 ⊢ (⟨x, z⟩ ∈ I ↔ z = x)
15	14	anbi1i 483	. . . . . . . . . 10 ⊢ ((⟨x, z⟩ ∈ I ⋀ x ∈ A) ↔ (z = x ⋀ x ∈ A))
16	10, 15	bitr 173	. . . . . . . . 9 ⊢ (⟨x, z⟩ ∈ (I ↾ A) ↔ (z = x ⋀ x ∈ A))
17	16	anbi1i 483	. . . . . . . 8 ⊢ ((⟨x, z⟩ ∈ (I ↾ A) ⋀ ⟨z, y⟩ ∈ F) ↔ ((z = x ⋀ x ∈ A) ⋀ ⟨z, y⟩ ∈ F))
18		anass 441	. . . . . . . 8 ⊢ (((z = x ⋀ x ∈ A) ⋀ ⟨z, y⟩ ∈ F) ↔ (z = x ⋀ (x ∈ A ⋀ ⟨z, y⟩ ∈ F)))
19	17, 18	bitr 173	. . . . . . 7 ⊢ ((⟨x, z⟩ ∈ (I ↾ A) ⋀ ⟨z, y⟩ ∈ F) ↔ (z = x ⋀ (x ∈ A ⋀ ⟨z, y⟩ ∈ F)))
20	19	exbii 1053	. . . . . 6 ⊢ (∃z(⟨x, z⟩ ∈ (I ↾ A) ⋀ ⟨z, y⟩ ∈ F) ↔ ∃z(z = x ⋀ (x ∈ A ⋀ ⟨z, y⟩ ∈ F)))
21		opeq1 2491	. . . . . . . . 9 ⊢ (z = x → ⟨z, y⟩ = ⟨x, y⟩)
22	21	eleq1d 1543	. . . . . . . 8 ⊢ (z = x → (⟨z, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ F))
23	22	anbi2d 618	. . . . . . 7 ⊢ (z = x → ((x ∈ A ⋀ ⟨z, y⟩ ∈ F) ↔ (x ∈ A ⋀ ⟨x, y⟩ ∈ F)))
24	5, 23	ceqsexv 1838	. . . . . 6 ⊢ (∃z(z = x ⋀ (x ∈ A ⋀ ⟨z, y⟩ ∈ F)) ↔ (x ∈ A ⋀ ⟨x, y⟩ ∈ F))
25	8, 20, 24	3bitr 177	. . . . 5 ⊢ (⟨x, y⟩ ∈ (F ∘ (I ↾ A)) ↔ (x ∈ A ⋀ ⟨x, y⟩ ∈ F))
26	4, 25	syl6rbbr 541	. . . 4 ⊢ (F Fn A → (⟨x, y⟩ ∈ (F ∘ (I ↾ A)) ↔ ⟨x, y⟩ ∈ F))
27	26	19.21aivv 1289	. . 3 ⊢ (F Fn A → ∀x∀y(⟨x, y⟩ ∈ (F ∘ (I ↾ A)) ↔ ⟨x, y⟩ ∈ F))
28		fnrel 3592	. . . 4 ⊢ (F Fn A → Rel F)
29		relco 3490	. . . . 5 ⊢ Rel (F ∘ (I ↾ A))
30		eqrel 3256	. . . . 5 ⊢ ((Rel (F ∘ (I ↾ A)) ⋀ Rel F) → ((F ∘ (I ↾ A)) = F ↔ ∀x∀y(⟨x, y⟩ ∈ (F ∘ (I ↾ A)) ↔ ⟨x, y⟩ ∈ F)))
31	29, 30	mpan 697	. . . 4 ⊢ (Rel F → ((F ∘ (I ↾ A)) = F ↔ ∀x∀y(⟨x, y⟩ ∈ (F ∘ (I ↾ A)) ↔ ⟨x, y⟩ ∈ F)))
32	28, 31	syl 10	. . 3 ⊢ (F Fn A → ((F ∘ (I ↾ A)) = F ↔ ∀x∀y(⟨x, y⟩ ∈ (F ∘ (I ↾ A)) ↔ ⟨x, y⟩ ∈ F)))
33	27, 32	mpbird 196	. 2 ⊢ (F Fn A → (F ∘ (I ↾ A)) = F)
34	1, 33	syl 10	1 ⊢ (F:A–→B → (F ∘ (I ↾ A)) = F)

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 956   = wceq 958   ∈ wcel 960  ∃wex 982  Vcvv 1814  ⟨cop 2415   class class class wbr 2624  Icid 2837   ↾ cres 3178   ∘ ccom 3180  Rel wrel 3181   Fn wfn 3183  –→wf 3184

This theorem is referenced by:  mapenlem1 4495  mapenlem2 4496  hoico1t 9677  hmeogrp 10524

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-co 3193  df-dm 3194  df-res 3196  df-fun 3198  df-fn 3199  df-f 3200

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