Theorem fcnvres 5314

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 4925	. 2 ⊢ Rel ◡(𝐹 ↾ 𝐴)
2		relres 4855	. 2 ⊢ Rel (◡𝐹 ↾ 𝐵)
3		opelf 5302	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
4	3	simpld 111	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑥 ∈ 𝐴)
5	4	ex 114	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑥 ∈ 𝐴))
6	5	pm4.71d 391	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ 𝑥 ∈ 𝐴)))
7		vex 2692	. . . . . 6 ⊢ 𝑦 ∈ V
8		vex 2692	. . . . . 6 ⊢ 𝑥 ∈ V
9	7, 8	opelcnv 4729	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ 𝐴))
10	7	opelres 4832	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ 𝑥 ∈ 𝐴))
11	9, 10	bitri 183	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ 𝑥 ∈ 𝐴))
12	6, 11	syl6bbr 197	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡(𝐹 ↾ 𝐴)))
13	3	simprd 113	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ 𝐵)
14	13	ex 114	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑦 ∈ 𝐵))
15	14	pm4.71d 391	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ 𝑦 ∈ 𝐵)))
16	8	opelres 4832	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ (◡𝐹 ↾ 𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ ◡𝐹 ∧ 𝑦 ∈ 𝐵))
17	7, 8	opelcnv 4729	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
18	17	anbi1i 454	. . . . 5 ⊢ ((⟨𝑦, 𝑥⟩ ∈ ◡𝐹 ∧ 𝑦 ∈ 𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ 𝑦 ∈ 𝐵))
19	16, 18	bitri 183	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ (◡𝐹 ↾ 𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ 𝑦 ∈ 𝐵))
20	15, 19	syl6bbr 197	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ (◡𝐹 ↾ 𝐵)))
21	12, 20	bitr3d 189	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑦, 𝑥⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ (◡𝐹 ↾ 𝐵)))
22	1, 2, 21	eqrelrdv 4643	1 ⊢ (𝐹:𝐴⟶𝐵 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481  ⟨cop 3535  ^◡ccnv 4546   ↾ cres 4549  ⟶wf 5127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-fun 5133  df-fn 5134  df-f 5135

This theorem is referenced by: (None)