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Theorem cnvssco 37379
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 2039 . 2 (∀𝑦𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)) ↔ ∀𝑥𝑦(⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)))
2 relcnv 5466 . . 3 Rel 𝐴
3 ssrel 5173 . . 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 1912 . . . 4 (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧𝑧𝐵𝑦)) ↔ (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑦)))
6 vex 3194 . . . . . . 7 𝑦 ∈ V
7 vex 3194 . . . . . . 7 𝑥 ∈ V
86, 7brcnv 5270 . . . . . 6 (𝑦𝐴𝑥𝑥𝐴𝑦)
9 df-br 4619 . . . . . 6 (𝑦𝐴𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
108, 9bitr3i 266 . . . . 5 (𝑥𝐴𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
117, 6brco 5257 . . . . . 6 (𝑥(𝐵𝐶)𝑦 ↔ ∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑦))
126, 7brcnv 5270 . . . . . . 7 (𝑦(𝐵𝐶)𝑥𝑥(𝐵𝐶)𝑦)
13 df-br 4619 . . . . . . 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 340 . . . 4 ((𝑥𝐴𝑦 → ∃𝑧(𝑥𝐶𝑧𝑧𝐵𝑦)) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)))
175, 16bitri 264 . . 3 (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧𝑧𝐵𝑦)) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ (𝐵𝐶)))
18172albii 1745 . 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 384  wal 1478  wex 1701  wcel 1992  wss 3560  cop 4159   class class class wbr 4618  ccnv 5078  ccom 5083  Rel wrel 5084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088
This theorem is referenced by:  refimssco  37380
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