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Theorem cnvssco 38229
 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 2077 . 2 (∀𝑦𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)) ↔ ∀𝑥𝑦(⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)))
2 relcnv 5538 . . 3 Rel 𝐴
3 ssrel 5241 . . 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 1966 . . . 4 (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧𝑧𝐵𝑦)) ↔ (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑦)))
6 vex 3234 . . . . . . 7 𝑦 ∈ V
7 vex 3234 . . . . . . 7 𝑥 ∈ V
86, 7brcnv 5337 . . . . . 6 (𝑦𝐴𝑥𝑥𝐴𝑦)
9 df-br 4686 . . . . . 6 (𝑦𝐴𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
108, 9bitr3i 266 . . . . 5 (𝑥𝐴𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
117, 6brco 5325 . . . . . 6 (𝑥(𝐵𝐶)𝑦 ↔ ∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑦))
126, 7brcnv 5337 . . . . . . 7 (𝑦(𝐵𝐶)𝑥𝑥(𝐵𝐶)𝑦)
13 df-br 4686 . . . . . . 7 (𝑦(𝐵𝐶)𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶))
1412, 13bitr3i 266 . . . . . 6 (𝑥(𝐵𝐶)𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶))
1511, 14bitr3i 266 . . . . 5 (∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑦) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶))
1610, 15imbi12i 339 . . . 4 ((𝑥𝐴𝑦 → ∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑦)) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)))
175, 16bitri 264 . . 3 (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧𝑧𝐵𝑦)) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)))
18172albii 1788 . 2 (∀𝑥𝑦𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧𝑧𝐵𝑦)) ↔ ∀𝑥𝑦(⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)))
191, 4, 183bitr4i 292 1 (𝐴(𝐵𝐶) ↔ ∀𝑥𝑦𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧𝑧𝐵𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1521  ∃wex 1744   ∈ wcel 2030   ⊆ wss 3607  ⟨cop 4216   class class class wbr 4685  ◡ccnv 5142   ∘ ccom 5147  Rel wrel 5148 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152 This theorem is referenced by:  refimssco  38230
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