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Theorem cnvssco 43596
Description: A condition weaker than reflexivity. (Contributed by RP, 3-Aug-2020.)
Assertion
Ref Expression
cnvssco (𝐴(𝐵𝐶) ↔ ∀𝑥𝑦𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧𝑧𝐵𝑦)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧

Proof of Theorem cnvssco
StepHypRef Expression
1 alcom 2157 . 2 (∀𝑦𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)) ↔ ∀𝑥𝑦(⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)))
2 relcnv 6125 . . 3 Rel 𝐴
3 ssrel 5795 . . 3 (Rel 𝐴 → (𝐴(𝐵𝐶) ↔ ∀𝑦𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶))))
42, 3ax-mp 5 . 2 (𝐴(𝐵𝐶) ↔ ∀𝑦𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)))
5 19.37v 1989 . . . 4 (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧𝑧𝐵𝑦)) ↔ (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑦)))
6 vex 3482 . . . . . . 7 𝑦 ∈ V
7 vex 3482 . . . . . . 7 𝑥 ∈ V
86, 7brcnv 5896 . . . . . 6 (𝑦𝐴𝑥𝑥𝐴𝑦)
9 df-br 5149 . . . . . 6 (𝑦𝐴𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
108, 9bitr3i 277 . . . . 5 (𝑥𝐴𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
117, 6brco 5884 . . . . . 6 (𝑥(𝐵𝐶)𝑦 ↔ ∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑦))
126, 7brcnv 5896 . . . . . . 7 (𝑦(𝐵𝐶)𝑥𝑥(𝐵𝐶)𝑦)
13 df-br 5149 . . . . . . 7 (𝑦(𝐵𝐶)𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶))
1412, 13bitr3i 277 . . . . . 6 (𝑥(𝐵𝐶)𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶))
1511, 14bitr3i 277 . . . . 5 (∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑦) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶))
1610, 15imbi12i 350 . . . 4 ((𝑥𝐴𝑦 → ∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑦)) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)))
175, 16bitri 275 . . 3 (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧𝑧𝐵𝑦)) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)))
18172albii 1817 . 2 (∀𝑥𝑦𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧𝑧𝐵𝑦)) ↔ ∀𝑥𝑦(⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)))
191, 4, 183bitr4i 303 1 (𝐴(𝐵𝐶) ↔ ∀𝑥𝑦𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧𝑧𝐵𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535  wex 1776  wcel 2106  wss 3963  cop 4637   class class class wbr 5148  ccnv 5688  ccom 5693  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698
This theorem is referenced by:  refimssco  43597
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