Theorem fcnvres 6786

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 6125	. 2 ⊢ Rel ◡(𝐹 ↾ 𝐴)
2		relres 6026	. 2 ⊢ Rel (◡𝐹 ↾ 𝐵)
3		opelf 6770	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
4	3	simpld 494	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑥 ∈ 𝐴)
5	4	ex 412	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑥 ∈ 𝐴))
6	5	pm4.71rd 562	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
7		vex 3482	. . . . . 6 ⊢ 𝑦 ∈ V
8		vex 3482	. . . . . 6 ⊢ 𝑥 ∈ V
9	7, 8	opelcnv 5895	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ 𝐴))
10	7	opelresi 6008	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
11	9, 10	bitri 275	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
12	6, 11	bitr4di 289	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡(𝐹 ↾ 𝐴)))
13	3	simprd 495	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ 𝐵)
14	13	ex 412	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑦 ∈ 𝐵))
15	14	pm4.71rd 562	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (𝑦 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
16	8	opelresi 6008	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ (◡𝐹 ↾ 𝐵) ↔ (𝑦 ∈ 𝐵 ∧ ⟨𝑦, 𝑥⟩ ∈ ◡𝐹))
17	7, 8	opelcnv 5895	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
18	17	anbi2i 623	. . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ ⟨𝑦, 𝑥⟩ ∈ ◡𝐹) ↔ (𝑦 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
19	16, 18	bitri 275	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ (◡𝐹 ↾ 𝐵) ↔ (𝑦 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
20	15, 19	bitr4di 289	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ (◡𝐹 ↾ 𝐵)))
21	12, 20	bitr3d 281	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑦, 𝑥⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ (◡𝐹 ↾ 𝐵)))
22	1, 2, 21	eqrelrdv 5805	1 ⊢ (𝐹:𝐴⟶𝐵 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ⟨cop 4637  ^◡ccnv 5688   ↾ cres 5691  ⟶wf 6559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-fun 6565  df-fn 6566  df-f 6567

This theorem is referenced by: (None)