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Theorem fcnvres 6720
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 6057 . 2 Rel (𝐹𝐴)
2 relres 5967 . 2 Rel (𝐹𝐵)
3 opelf 6704 . . . . . . 7 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥𝐴𝑦𝐵))
43simpld 496 . . . . . 6 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑥𝐴)
54ex 414 . . . . 5 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥𝐴))
65pm4.71rd 564 . . . 4 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
7 vex 3450 . . . . . 6 𝑦 ∈ V
8 vex 3450 . . . . . 6 𝑥 ∈ V
97, 8opelcnv 5838 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐹𝐴))
107opelresi 5946 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐹𝐴) ↔ (𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
119, 10bitri 275 . . . 4 (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐴) ↔ (𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
126, 11bitr4di 289 . . 3 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐹𝐴)))
133simprd 497 . . . . . 6 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦𝐵)
1413ex 414 . . . . 5 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦𝐵))
1514pm4.71rd 564 . . . 4 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (𝑦𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
168opelresi 5946 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐵) ↔ (𝑦𝐵 ∧ ⟨𝑦, 𝑥⟩ ∈ 𝐹))
177, 8opelcnv 5838 . . . . . 6 (⟨𝑦, 𝑥⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
1817anbi2i 624 . . . . 5 ((𝑦𝐵 ∧ ⟨𝑦, 𝑥⟩ ∈ 𝐹) ↔ (𝑦𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
1916, 18bitri 275 . . . 4 (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐵) ↔ (𝑦𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
2015, 19bitr4di 289 . . 3 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐹𝐵)))
2112, 20bitr3d 281 . 2 (𝐹:𝐴𝐵 → (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐹𝐵)))
221, 2, 21eqrelrdv 5749 1 (𝐹:𝐴𝐵(𝐹𝐴) = (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  cop 4593  ccnv 5633  cres 5636  wf 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-fun 6499  df-fn 6500  df-f 6501
This theorem is referenced by: (None)
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