Theorem fcnvres 6756

Description: The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)

Assertion

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

Proof of Theorem fcnvres

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

Step	Hyp	Ref	Expression
1		relcnv 6107	. 2 ⊢ Rel ◡(𝐹 ↾ 𝐴)
2		relres 6005	. 2 ⊢ Rel (◡𝐹 ↾ 𝐵)
3		opelf 6740	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
4	3	simpld 499	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑥 ∈ 𝐴)
5	4	ex 417	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑥 ∈ 𝐴))
6	5	pm4.71rd 571	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
7		vex 3465	. . . . . 6 ⊢ 𝑦 ∈ V
8		vex 3465	. . . . . 6 ⊢ 𝑥 ∈ V
9	7, 8	opelcnv 5868	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ 𝐴))
10	7	opelresi 5987	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
11	9, 10	bitri 278	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
12	6, 11	bitr4di 292	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡(𝐹 ↾ 𝐴)))
13	3	simprd 500	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ 𝐵)
14	13	ex 417	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑦 ∈ 𝐵))
15	14	pm4.71rd 571	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (𝑦 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
16	8	opelresi 5987	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ (◡𝐹 ↾ 𝐵) ↔ (𝑦 ∈ 𝐵 ∧ ⟨𝑦, 𝑥⟩ ∈ ◡𝐹))
17	7, 8	opelcnv 5868	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
18	17	anbi2i 634	. . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ ⟨𝑦, 𝑥⟩ ∈ ◡𝐹) ↔ (𝑦 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
19	16, 18	bitri 278	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ (◡𝐹 ↾ 𝐵) ↔ (𝑦 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
20	15, 19	bitr4di 292	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ (◡𝐹 ↾ 𝐵)))
21	12, 20	bitr3d 284	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑦, 𝑥⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ (◡𝐹 ↾ 𝐵)))
22	1, 2, 21	eqrelrdv 5779	1 ⊢ (𝐹:𝐴⟶𝐵 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ 𝐵))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ⟨cop 4598  ^◡ccnv 5661   ↾ cres 5664  ⟶wf 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5259  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-br 5112  df-opab 5176  df-xp 5668  df-rel 5669  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-fun 6539  df-fn 6540  df-f 6541

This theorem is referenced by: (None)