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Theorem fcnvres 5194
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 4810 . 2 Rel (𝐹𝐴)
2 relres 4741 . 2 Rel (𝐹𝐵)
3 opelf 5182 . . . . . . 7 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥𝐴𝑦𝐵))
43simpld 110 . . . . . 6 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑥𝐴)
54ex 113 . . . . 5 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥𝐴))
65pm4.71d 385 . . . 4 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥𝐴)))
7 vex 2622 . . . . . 6 𝑦 ∈ V
8 vex 2622 . . . . . 6 𝑥 ∈ V
97, 8opelcnv 4618 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐹𝐴))
107opelres 4718 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐹𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥𝐴))
119, 10bitri 182 . . . 4 (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥𝐴))
126, 11syl6bbr 196 . . 3 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐹𝐴)))
133simprd 112 . . . . . 6 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦𝐵)
1413ex 113 . . . . 5 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦𝐵))
1514pm4.71d 385 . . . 4 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦𝐵)))
168opelres 4718 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐹𝑦𝐵))
177, 8opelcnv 4618 . . . . . 6 (⟨𝑦, 𝑥⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
1817anbi1i 446 . . . . 5 ((⟨𝑦, 𝑥⟩ ∈ 𝐹𝑦𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦𝐵))
1916, 18bitri 182 . . . 4 (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦𝐵))
2015, 19syl6bbr 196 . . 3 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐹𝐵)))
2112, 20bitr3d 188 . 2 (𝐹:𝐴𝐵 → (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐹𝐵)))
221, 2, 21eqrelrdv 4534 1 (𝐹:𝐴𝐵(𝐹𝐴) = (𝐹𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  cop 3449  ccnv 4437  cres 4440  wf 5011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-fun 5017  df-fn 5018  df-f 5019
This theorem is referenced by: (None)
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