Theorem fcnvres 5207

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 4823	. 2 ⊢ Rel ◡(𝐹 ↾ 𝐴)
2		relres 4754	. 2 ⊢ Rel (◡𝐹 ↾ 𝐵)
3		opelf 5195	. . . . . . 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 386	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ 𝑥 ∈ 𝐴)))
7		vex 2623	. . . . . 6 ⊢ 𝑦 ∈ V
8		vex 2623	. . . . . 6 ⊢ 𝑥 ∈ V
9	7, 8	opelcnv 4631	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ 𝐴))
10	7	opelres 4731	. . . . 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 386	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ 𝑦 ∈ 𝐵)))
16	8	opelres 4731	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ (◡𝐹 ↾ 𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ ◡𝐹 ∧ 𝑦 ∈ 𝐵))
17	7, 8	opelcnv 4631	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
18	17	anbi1i 447	. . . . 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 4547	1 ⊢ (𝐹:𝐴⟶𝐵 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1290   ∈ wcel 1439  ⟨cop 3453  ^◡ccnv 4451   ↾ cres 4454  ⟶wf 5024

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4458  df-rel 4459  df-cnv 4460  df-dm 4462  df-rn 4463  df-res 4464  df-fun 5030  df-fn 5031  df-f 5032

This theorem is referenced by: (None)