Theorem fcnvres 2768

Description: The converse of a restriction of a function.

Assertion

Ref	Expression
fcnvres	⊢ (F:A–→B → ◡(F ↾ A) = (◡F ↾ B))

Proof of Theorem fcnvres

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . . . . 9 ⊢ y ∈ V
2	1	opelf 2762	. . . . . . . 8 ⊢ ((F:A–→B ∧ ⟨x, y⟩ ∈ F) → (x ∈ A ∧ y ∈ B))
3	2	pm3.26d 258	. . . . . . 7 ⊢ ((F:A–→B ∧ ⟨x, y⟩ ∈ F) → x ∈ A)
4	3	exp 291	. . . . . 6 ⊢ (F:A–→B → (⟨x, y⟩ ∈ F → x ∈ A))
5		pm4.71 481	. . . . . 6 ⊢ ((⟨x, y⟩ ∈ F → x ∈ A) ↔ (⟨x, y⟩ ∈ F ↔ (⟨x, y⟩ ∈ F ∧ x ∈ A)))
6	4, 5	sylib 173	. . . . 5 ⊢ (F:A–→B → (⟨x, y⟩ ∈ F ↔ (⟨x, y⟩ ∈ F ∧ x ∈ A)))
7		visset 1350	. . . . . . 7 ⊢ x ∈ V
8	1, 7	opelcnv 2518	. . . . . 6 ⊢ (⟨y, x⟩ ∈ ◡(F ↾ A) ↔ ⟨x, y⟩ ∈ (F ↾ A))
9	1	opelres 2579	. . . . . 6 ⊢ (⟨x, y⟩ ∈ (F ↾ A) ↔ (⟨x, y⟩ ∈ F ∧ x ∈ A))
10	8, 9	bitr 151	. . . . 5 ⊢ (⟨y, x⟩ ∈ ◡(F ↾ A) ↔ (⟨x, y⟩ ∈ F ∧ x ∈ A))
11	6, 10	syl6bbr 416	. . . 4 ⊢ (F:A–→B → (⟨x, y⟩ ∈ F ↔ ⟨y, x⟩ ∈ ◡(F ↾ A)))
12	2	pm3.27d 262	. . . . . . 7 ⊢ ((F:A–→B ∧ ⟨x, y⟩ ∈ F) → y ∈ B)
13	12	exp 291	. . . . . 6 ⊢ (F:A–→B → (⟨x, y⟩ ∈ F → y ∈ B))
14		pm4.71 481	. . . . . 6 ⊢ ((⟨x, y⟩ ∈ F → y ∈ B) ↔ (⟨x, y⟩ ∈ F ↔ (⟨x, y⟩ ∈ F ∧ y ∈ B)))
15	13, 14	sylib 173	. . . . 5 ⊢ (F:A–→B → (⟨x, y⟩ ∈ F ↔ (⟨x, y⟩ ∈ F ∧ y ∈ B)))
16	7	opelres 2579	. . . . . 6 ⊢ (⟨y, x⟩ ∈ (◡F ↾ B) ↔ (⟨y, x⟩ ∈ ◡F ∧ y ∈ B))
17	1, 7	opelcnv 2518	. . . . . . 7 ⊢ (⟨y, x⟩ ∈ ◡F ↔ ⟨x, y⟩ ∈ F)
18	17	anbi1i 368	. . . . . 6 ⊢ ((⟨y, x⟩ ∈ ◡F ∧ y ∈ B) ↔ (⟨x, y⟩ ∈ F ∧ y ∈ B))
19	16, 18	bitr 151	. . . . 5 ⊢ (⟨y, x⟩ ∈ (◡F ↾ B) ↔ (⟨x, y⟩ ∈ F ∧ y ∈ B))
20	15, 19	syl6bbr 416	. . . 4 ⊢ (F:A–→B → (⟨x, y⟩ ∈ F ↔ ⟨y, x⟩ ∈ (◡F ↾ B)))
21	11, 20	bitr3d 408	. . 3 ⊢ (F:A–→B → (⟨y, x⟩ ∈ ◡(F ↾ A) ↔ ⟨y, x⟩ ∈ (◡F ↾ B)))
22	21	19.21aivv 944	. 2 ⊢ (F:A–→B → ∀y∀x(⟨y, x⟩ ∈ ◡(F ↾ A) ↔ ⟨y, x⟩ ∈ (◡F ↾ B)))
23		relcnv 2624	. . 3 ⊢ Rel ◡(F ↾ A)
24		relres 2591	. . 3 ⊢ Rel (◡F ↾ B)
25		cleqrel 2483	. . 3 ⊢ ((Rel ◡(F ↾ A) ∧ Rel (◡F ↾ B)) → (◡(F ↾ A) = (◡F ↾ B) ↔ ∀y∀x(⟨y, x⟩ ∈ ◡(F ↾ A) ↔ ⟨y, x⟩ ∈ (◡F ↾ B))))
26	23, 24, 25	mp2an 520	. 2 ⊢ (◡(F ↾ A) = (◡F ↾ B) ↔ ∀y∀x(⟨y, x⟩ ∈ ◡(F ↾ A) ↔ ⟨y, x⟩ ∈ (◡F ↾ B)))
27	22, 26	sylibr 175	1 ⊢ (F:A–→B → ◡(F ↾ A) = (◡F ↾ B))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  ^◡ccnv 2409   ↾ cres 2412  Rel wrel 2415  –→wf 2418

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-dm 2428  df-rn 2429  df-res 2430  df-fun 2432  df-fn 2433  df-f 2434

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