Theorem fcnvres 5073

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 4703	. 2 ⊢ Rel ◡(𝐹 ↾ 𝐴)
2		relres 4639	. 2 ⊢ Rel (◡𝐹 ↾ 𝐵)
3		opelf 5062	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
4	3	simpld 105	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑥 ∈ 𝐴)
5	4	ex 108	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑥 ∈ 𝐴))
6	5	pm4.71d 373	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ 𝑥 ∈ 𝐴)))
7		vex 2560	. . . . . 6 ⊢ 𝑦 ∈ V
8		vex 2560	. . . . . 6 ⊢ 𝑥 ∈ V
9	7, 8	opelcnv 4517	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ 𝐴))
10	7	opelres 4617	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ 𝑥 ∈ 𝐴))
11	9, 10	bitri 173	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ 𝑥 ∈ 𝐴))
12	6, 11	syl6bbr 187	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡(𝐹 ↾ 𝐴)))
13	3	simprd 107	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦 ∈ 𝐵)
14	13	ex 108	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑦 ∈ 𝐵))
15	14	pm4.71d 373	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ 𝑦 ∈ 𝐵)))
16	8	opelres 4617	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ (◡𝐹 ↾ 𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ ◡𝐹 ∧ 𝑦 ∈ 𝐵))
17	7, 8	opelcnv 4517	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
18	17	anbi1i 431	. . . . 5 ⊢ ((⟨𝑦, 𝑥⟩ ∈ ◡𝐹 ∧ 𝑦 ∈ 𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ 𝑦 ∈ 𝐵))
19	16, 18	bitri 173	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ (◡𝐹 ↾ 𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ 𝑦 ∈ 𝐵))
20	15, 19	syl6bbr 187	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ (◡𝐹 ↾ 𝐵)))
21	12, 20	bitr3d 179	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (⟨𝑦, 𝑥⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ (◡𝐹 ↾ 𝐵)))
22	1, 2, 21	eqrelrdv 4436	1 ⊢ (𝐹:𝐴⟶𝐵 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ 𝐵))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  ⟨cop 3378  ^◡ccnv 4344   ↾ cres 4347  ⟶wf 4898

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-fun 4904  df-fn 4905  df-f 4906

This theorem is referenced by: (None)