Theorem cnvssco 44223

Description: A condition weaker than reflexivity. (Contributed by RP, 3-Aug-2020.)

Assertion

Ref	Expression
cnvssco	⊢ (◡𝐴 ⊆ ◡(𝐵 ∘ 𝐶) ↔ ∀𝑥∀𝑦∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧

Proof of Theorem cnvssco

Step	Hyp	Ref	Expression
1		alcom 2200	. 2 ⊢ (∀𝑦∀𝑥(⟨𝑦, 𝑥⟩ ∈ ◡𝐴 → ⟨𝑦, 𝑥⟩ ∈ ◡(𝐵 ∘ 𝐶)) ↔ ∀𝑥∀𝑦(⟨𝑦, 𝑥⟩ ∈ ◡𝐴 → ⟨𝑦, 𝑥⟩ ∈ ◡(𝐵 ∘ 𝐶)))
2		relcnv 6107	. . 3 ⊢ Rel ◡𝐴
3		ssrel 5770	. . 3 ⊢ (Rel ◡𝐴 → (◡𝐴 ⊆ ◡(𝐵 ∘ 𝐶) ↔ ∀𝑦∀𝑥(⟨𝑦, 𝑥⟩ ∈ ◡𝐴 → ⟨𝑦, 𝑥⟩ ∈ ◡(𝐵 ∘ 𝐶))))
4	2, 3	ax-mp 5	. 2 ⊢ (◡𝐴 ⊆ ◡(𝐵 ∘ 𝐶) ↔ ∀𝑦∀𝑥(⟨𝑦, 𝑥⟩ ∈ ◡𝐴 → ⟨𝑦, 𝑥⟩ ∈ ◡(𝐵 ∘ 𝐶)))
5		19.37v 2024	. . . 4 ⊢ (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)) ↔ (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)))
6		vex 3467	. . . . . . 7 ⊢ 𝑦 ∈ V
7		vex 3467	. . . . . . 7 ⊢ 𝑥 ∈ V
8	6, 7	brcnv 5869	. . . . . 6 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
9		df-br 5114	. . . . . 6 ⊢ (𝑦◡𝐴𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡𝐴)
10	8, 9	bitr3i 280	. . . . 5 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡𝐴)
11	7, 6	brco 5857	. . . . . 6 ⊢ (𝑥(𝐵 ∘ 𝐶)𝑦 ↔ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦))
12	6, 7	brcnv 5869	. . . . . . 7 ⊢ (𝑦◡(𝐵 ∘ 𝐶)𝑥 ↔ 𝑥(𝐵 ∘ 𝐶)𝑦)
13		df-br 5114	. . . . . . 7 ⊢ (𝑦◡(𝐵 ∘ 𝐶)𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡(𝐵 ∘ 𝐶))
14	12, 13	bitr3i 280	. . . . . 6 ⊢ (𝑥(𝐵 ∘ 𝐶)𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡(𝐵 ∘ 𝐶))
15	11, 14	bitr3i 280	. . . . 5 ⊢ (∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦) ↔ ⟨𝑦, 𝑥⟩ ∈ ◡(𝐵 ∘ 𝐶))
16	10, 15	imbi12i 353	. . . 4 ⊢ ((𝑥𝐴𝑦 → ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)) ↔ (⟨𝑦, 𝑥⟩ ∈ ◡𝐴 → ⟨𝑦, 𝑥⟩ ∈ ◡(𝐵 ∘ 𝐶)))
17	5, 16	bitri 278	. . 3 ⊢ (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)) ↔ (⟨𝑦, 𝑥⟩ ∈ ◡𝐴 → ⟨𝑦, 𝑥⟩ ∈ ◡(𝐵 ∘ 𝐶)))
18	17	2albii 1847	. 2 ⊢ (∀𝑥∀𝑦∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)) ↔ ∀𝑥∀𝑦(⟨𝑦, 𝑥⟩ ∈ ◡𝐴 → ⟨𝑦, 𝑥⟩ ∈ ◡(𝐵 ∘ 𝐶)))
19	1, 4, 18	3bitr4i 306	1 ⊢ (◡𝐴 ⊆ ◡(𝐵 ∘ 𝐶) ↔ ∀𝑥∀𝑦∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565  ∃wex 1806   ∈ wcel 2149   ⊆ wss 3913  ⟨cop 4600   class class class wbr 5113  ^◡ccnv 5661   ∘ ccom 5666  Rel wrel 5667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671

This theorem is referenced by:  refimssco  44224