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Theorem fcnvres 6293
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 5713 . 2 Rel (𝐹𝐴)
2 relres 5629 . 2 Rel (𝐹𝐵)
3 opelf 6276 . . . . . . 7 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → (𝑥𝐴𝑦𝐵))
43simpld 484 . . . . . 6 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑥𝐴)
54ex 399 . . . . 5 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥𝐴))
65pm4.71d 553 . . . 4 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥𝐴)))
7 vex 3394 . . . . . 6 𝑦 ∈ V
8 vex 3394 . . . . . 6 𝑥 ∈ V
97, 8opelcnv 5505 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐹𝐴))
107opelres 5605 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐹𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥𝐴))
119, 10bitri 266 . . . 4 (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥𝐴))
126, 11syl6bbr 280 . . 3 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐹𝐴)))
133simprd 485 . . . . . 6 ((𝐹:𝐴𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) → 𝑦𝐵)
1413ex 399 . . . . 5 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦𝐵))
1514pm4.71d 553 . . . 4 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦𝐵)))
168opelres 5605 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐹𝑦𝐵))
177, 8opelcnv 5505 . . . . . 6 (⟨𝑦, 𝑥⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
1817anbi1i 612 . . . . 5 ((⟨𝑦, 𝑥⟩ ∈ 𝐹𝑦𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦𝐵))
1916, 18bitri 266 . . . 4 (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦𝐵))
2015, 19syl6bbr 280 . . 3 (𝐹:𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐹𝐵)))
2112, 20bitr3d 272 . 2 (𝐹:𝐴𝐵 → (⟨𝑦, 𝑥⟩ ∈ (𝐹𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐹𝐵)))
221, 2, 21eqrelrdv 5418 1 (𝐹:𝐴𝐵(𝐹𝐴) = (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2156  cop 4376  ccnv 5310  cres 5313  wf 6093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-br 4845  df-opab 4907  df-xp 5317  df-rel 5318  df-cnv 5319  df-dm 5321  df-rn 5322  df-res 5323  df-fun 6099  df-fn 6100  df-f 6101
This theorem is referenced by: (None)
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