Theorem cnvssco 43991

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 2165	. 2 ⊢ (∀𝑦∀𝑥(⟨𝑦, 𝑥⟩ ∈ ◡𝐴 → ⟨𝑦, 𝑥⟩ ∈ ◡(𝐵 ∘ 𝐶)) ↔ ∀𝑥∀𝑦(⟨𝑦, 𝑥⟩ ∈ ◡𝐴 → ⟨𝑦, 𝑥⟩ ∈ ◡(𝐵 ∘ 𝐶)))
2		relcnv 6073	. . 3 ⊢ Rel ◡𝐴
3		ssrel 5742	. . 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 1999	. . . 4 ⊢ (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)) ↔ (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)))
6		vex 3446	. . . . . . 7 ⊢ 𝑦 ∈ V
7		vex 3446	. . . . . . 7 ⊢ 𝑥 ∈ V
8	6, 7	brcnv 5841	. . . . . 6 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
9		df-br 5101	. . . . . 6 ⊢ (𝑦◡𝐴𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡𝐴)
10	8, 9	bitr3i 277	. . . . 5 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡𝐴)
11	7, 6	brco 5829	. . . . . 6 ⊢ (𝑥(𝐵 ∘ 𝐶)𝑦 ↔ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦))
12	6, 7	brcnv 5841	. . . . . . 7 ⊢ (𝑦◡(𝐵 ∘ 𝐶)𝑥 ↔ 𝑥(𝐵 ∘ 𝐶)𝑦)
13		df-br 5101	. . . . . . 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 1822	. 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 1540  ∃wex 1781   ∈ wcel 2114   ⊆ wss 3903  ⟨cop 4588   class class class wbr 5100  ^◡ccnv 5633   ∘ ccom 5638  Rel wrel 5639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5245  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643

This theorem is referenced by:  refimssco  43992