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Theorem cnvssco 38587
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 2201 . 2 (∀𝑦𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)) ↔ ∀𝑥𝑦(⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)))
2 relcnv 5685 . . 3 Rel 𝐴
3 ssrel 5377 . . 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 2090 . . . 4 (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧𝑧𝐵𝑦)) ↔ (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑦)))
6 vex 3353 . . . . . . 7 𝑦 ∈ V
7 vex 3353 . . . . . . 7 𝑥 ∈ V
86, 7brcnv 5473 . . . . . 6 (𝑦𝐴𝑥𝑥𝐴𝑦)
9 df-br 4810 . . . . . 6 (𝑦𝐴𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
108, 9bitr3i 268 . . . . 5 (𝑥𝐴𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
117, 6brco 5461 . . . . . 6 (𝑥(𝐵𝐶)𝑦 ↔ ∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑦))
126, 7brcnv 5473 . . . . . . 7 (𝑦(𝐵𝐶)𝑥𝑥(𝐵𝐶)𝑦)
13 df-br 4810 . . . . . . 7 (𝑦(𝐵𝐶)𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶))
1412, 13bitr3i 268 . . . . . 6 (𝑥(𝐵𝐶)𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶))
1511, 14bitr3i 268 . . . . 5 (∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑦) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶))
1610, 15imbi12i 341 . . . 4 ((𝑥𝐴𝑦 → ∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑦)) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)))
175, 16bitri 266 . . 3 (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧𝑧𝐵𝑦)) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)))
18172albii 1915 . 2 (∀𝑥𝑦𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧𝑧𝐵𝑦)) ↔ ∀𝑥𝑦(⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)))
191, 4, 183bitr4i 294 1 (𝐴(𝐵𝐶) ↔ ∀𝑥𝑦𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧𝑧𝐵𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650  wex 1874  wcel 2155  wss 3732  cop 4340   class class class wbr 4809  ccnv 5276  ccom 5281  Rel wrel 5282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-opab 4872  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286
This theorem is referenced by:  refimssco  38588
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