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Theorem cnvssco 39844
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 2153 . 2 (∀𝑦𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)) ↔ ∀𝑥𝑦(⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)))
2 relcnv 5960 . . 3 Rel 𝐴
3 ssrel 5650 . . 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 3495 . . . . . . 7 𝑦 ∈ V
7 vex 3495 . . . . . . 7 𝑥 ∈ V
86, 7brcnv 5746 . . . . . 6 (𝑦𝐴𝑥𝑥𝐴𝑦)
9 df-br 5058 . . . . . 6 (𝑦𝐴𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
108, 9bitr3i 278 . . . . 5 (𝑥𝐴𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
117, 6brco 5734 . . . . . 6 (𝑥(𝐵𝐶)𝑦 ↔ ∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑦))
126, 7brcnv 5746 . . . . . . 7 (𝑦(𝐵𝐶)𝑥𝑥(𝐵𝐶)𝑦)
13 df-br 5058 . . . . . . 7 (𝑦(𝐵𝐶)𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶))
1412, 13bitr3i 278 . . . . . 6 (𝑥(𝐵𝐶)𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶))
1511, 14bitr3i 278 . . . . 5 (∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑦) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶))
1610, 15imbi12i 352 . . . 4 ((𝑥𝐴𝑦 → ∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑦)) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)))
175, 16bitri 276 . . 3 (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧𝑧𝐵𝑦)) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)))
18172albii 1812 . 2 (∀𝑥𝑦𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧𝑧𝐵𝑦)) ↔ ∀𝑥𝑦(⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)))
191, 4, 183bitr4i 304 1 (𝐴(𝐵𝐶) ↔ ∀𝑥𝑦𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧𝑧𝐵𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526  wex 1771  wcel 2105  wss 3933  cop 4563   class class class wbr 5057  ccnv 5547  ccom 5552  Rel wrel 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557
This theorem is referenced by:  refimssco  39845
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