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Theorem fcnvres 6531
 Description: The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
Assertion
Ref Expression
fcnvres (𝐹:𝐴𝐵(𝐹𝐴) = (𝐹𝐵))

Proof of Theorem fcnvres
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 5935 . 2 Rel (𝐹𝐴)
2 relres 5848 . 2 Rel (𝐹𝐵)
3 opelf 6514 . . . . . . 7 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥𝐴𝑦𝐵))
43simpld 498 . . . . . 6 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑥𝐴)
54ex 416 . . . . 5 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥𝐴))
65pm4.71rd 566 . . . 4 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
7 vex 3444 . . . . . 6 𝑦 ∈ V
8 vex 3444 . . . . . 6 𝑥 ∈ V
97, 8opelcnv 5717 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐹𝐴))
107opelresi 5827 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐹𝐴) ↔ (𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
119, 10bitri 278 . . . 4 (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐴) ↔ (𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
126, 11bitr4di 292 . . 3 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐹𝐴)))
133simprd 499 . . . . . 6 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦𝐵)
1413ex 416 . . . . 5 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦𝐵))
1514pm4.71rd 566 . . . 4 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (𝑦𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
168opelresi 5827 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐵) ↔ (𝑦𝐵 ∧ ⟨𝑦, 𝑥⟩ ∈ 𝐹))
177, 8opelcnv 5717 . . . . . 6 (⟨𝑦, 𝑥⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
1817anbi2i 625 . . . . 5 ((𝑦𝐵 ∧ ⟨𝑦, 𝑥⟩ ∈ 𝐹) ↔ (𝑦𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
1916, 18bitri 278 . . . 4 (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐵) ↔ (𝑦𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
2015, 19bitr4di 292 . . 3 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐹𝐵)))
2112, 20bitr3d 284 . 2 (𝐹:𝐴𝐵 → (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐹𝐵)))
221, 2, 21eqrelrdv 5630 1 (𝐹:𝐴𝐵(𝐹𝐴) = (𝐹𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ⟨cop 4531  ◡ccnv 5519   ↾ cres 5522  ⟶wf 6321 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-xp 5526  df-rel 5527  df-cnv 5528  df-dm 5530  df-rn 5531  df-res 5532  df-fun 6327  df-fn 6328  df-f 6329 This theorem is referenced by: (None)
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