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Theorem fcnvres 5980
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 5409 . 2 Rel (𝐹𝐴)
2 relres 5333 . 2 Rel (𝐹𝐵)
3 opelf 5964 . . . . . . 7 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥𝐴𝑦𝐵))
43simpld 474 . . . . . 6 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑥𝐴)
54ex 449 . . . . 5 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥𝐴))
65pm4.71d 664 . . . 4 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥𝐴)))
7 vex 3176 . . . . . 6 𝑦 ∈ V
8 vex 3176 . . . . . 6 𝑥 ∈ V
97, 8opelcnv 5214 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐹𝐴))
107opelres 5309 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐹𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥𝐴))
119, 10bitri 263 . . . 4 (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥𝐴))
126, 11syl6bbr 277 . . 3 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐹𝐴)))
133simprd 478 . . . . . 6 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦𝐵)
1413ex 449 . . . . 5 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦𝐵))
1514pm4.71d 664 . . . 4 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦𝐵)))
168opelres 5309 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐹𝑦𝐵))
177, 8opelcnv 5214 . . . . . 6 (⟨𝑦, 𝑥⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
1817anbi1i 727 . . . . 5 ((⟨𝑦, 𝑥⟩ ∈ 𝐹𝑦𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦𝐵))
1916, 18bitri 263 . . . 4 (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦𝐵))
2015, 19syl6bbr 277 . . 3 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐹𝐵)))
2112, 20bitr3d 269 . 2 (𝐹:𝐴𝐵 → (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐹𝐵)))
221, 2, 21eqrelrdv 5128 1 (𝐹:𝐴𝐵(𝐹𝐴) = (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  cop 4131  ccnv 5027  cres 5030  wf 5786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4579  df-opab 4639  df-xp 5034  df-rel 5035  df-cnv 5036  df-dm 5038  df-rn 5039  df-res 5040  df-fun 5792  df-fn 5793  df-f 5794
This theorem is referenced by: (None)
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