Theorem cnvssco 43589

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 2160	. 2 ⊢ (∀𝑦∀𝑥(⟨𝑦, 𝑥⟩ ∈ ◡𝐴 → ⟨𝑦, 𝑥⟩ ∈ ◡(𝐵 ∘ 𝐶)) ↔ ∀𝑥∀𝑦(⟨𝑦, 𝑥⟩ ∈ ◡𝐴 → ⟨𝑦, 𝑥⟩ ∈ ◡(𝐵 ∘ 𝐶)))
2		relcnv 6064	. . 3 ⊢ Rel ◡𝐴
3		ssrel 5737	. . 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 1997	. . . 4 ⊢ (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)) ↔ (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)))
6		vex 3448	. . . . . . 7 ⊢ 𝑦 ∈ V
7		vex 3448	. . . . . . 7 ⊢ 𝑥 ∈ V
8	6, 7	brcnv 5836	. . . . . 6 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
9		df-br 5103	. . . . . 6 ⊢ (𝑦◡𝐴𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡𝐴)
10	8, 9	bitr3i 277	. . . . 5 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡𝐴)
11	7, 6	brco 5824	. . . . . 6 ⊢ (𝑥(𝐵 ∘ 𝐶)𝑦 ↔ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦))
12	6, 7	brcnv 5836	. . . . . . 7 ⊢ (𝑦◡(𝐵 ∘ 𝐶)𝑥 ↔ 𝑥(𝐵 ∘ 𝐶)𝑦)
13		df-br 5103	. . . . . . 7 ⊢ (𝑦◡(𝐵 ∘ 𝐶)𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡(𝐵 ∘ 𝐶))
14	12, 13	bitr3i 277	. . . . . 6 ⊢ (𝑥(𝐵 ∘ 𝐶)𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡(𝐵 ∘ 𝐶))
15	11, 14	bitr3i 277	. . . . 5 ⊢ (∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦) ↔ ⟨𝑦, 𝑥⟩ ∈ ◡(𝐵 ∘ 𝐶))
16	10, 15	imbi12i 350	. . . 4 ⊢ ((𝑥𝐴𝑦 → ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)) ↔ (⟨𝑦, 𝑥⟩ ∈ ◡𝐴 → ⟨𝑦, 𝑥⟩ ∈ ◡(𝐵 ∘ 𝐶)))
17	5, 16	bitri 275	. . 3 ⊢ (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)) ↔ (⟨𝑦, 𝑥⟩ ∈ ◡𝐴 → ⟨𝑦, 𝑥⟩ ∈ ◡(𝐵 ∘ 𝐶)))
18	17	2albii 1820	. 2 ⊢ (∀𝑥∀𝑦∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)) ↔ ∀𝑥∀𝑦(⟨𝑦, 𝑥⟩ ∈ ◡𝐴 → ⟨𝑦, 𝑥⟩ ∈ ◡(𝐵 ∘ 𝐶)))
19	1, 4, 18	3bitr4i 303	1 ⊢ (◡𝐴 ⊆ ◡(𝐵 ∘ 𝐶) ↔ ∀𝑥∀𝑦∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538  ∃wex 1779   ∈ wcel 2109   ⊆ wss 3911  ⟨cop 4591   class class class wbr 5102  ^◡ccnv 5630   ∘ ccom 5635  Rel wrel 5636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640

This theorem is referenced by:  refimssco  43590