Theorem trcoss2 39073

Description: Equivalent expressions for the transitivity of cosets by 𝑅. (Contributed by Peter Mazsa, 4-Jul-2020.) (Revised by Peter Mazsa, 16-Oct-2021.)

Assertion

Ref	Expression
trcoss2	⊢ (∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧) ↔ ∀𝑥∀𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅))

Distinct variable groups:   𝑦,𝑅   𝑥,𝑦   𝑦,𝑧

Allowed substitution hints:   𝑅(𝑥,𝑧)

Proof of Theorem trcoss2

Step	Hyp	Ref	Expression
1		alcom 2193	. . 3 ⊢ (∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧) ↔ ∀𝑧∀𝑦((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧))
2	1	albii 1839	. 2 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧) ↔ ∀𝑥∀𝑧∀𝑦((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧))
3		19.23v 1962	. . . 4 ⊢ (∀𝑦(𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅) ↔ (∃𝑦 𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅))
4		eleccossin 39072	. . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ↔ (𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧)))
5	4	el2v 3461	. . . . . . 7 ⊢ (𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ↔ (𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧))
6	5	bicomi 226	. . . . . 6 ⊢ ((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) ↔ 𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅))
7		brcoss3 39022	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ≀ 𝑅𝑧 ↔ ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅))
8	7	el2v 3461	. . . . . 6 ⊢ (𝑥 ≀ 𝑅𝑧 ↔ ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅)
9	6, 8	imbi12i 352	. . . . 5 ⊢ (((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧) ↔ (𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅))
10	9	albii 1839	. . . 4 ⊢ (∀𝑦((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧) ↔ ∀𝑦(𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅))
11		n0 4305	. . . . 5 ⊢ (([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅))
12	11	imbi1i 351	. . . 4 ⊢ ((([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅) ↔ (∃𝑦 𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅))
13	3, 10, 12	3bitr4i 305	. . 3 ⊢ (∀𝑦((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧) ↔ (([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅))
14	13	2albii 1840	. 2 ⊢ (∀𝑥∀𝑧∀𝑦((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧) ↔ ∀𝑥∀𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅))
15	2, 14	bitri 277	1 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧) ↔ ∀𝑥∀𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1558  ∃wex 1799   ∈ wcel 2142   ≠ wne 2957  Vcvv 3454   ∩ cin 3903  ∅c0 4285   class class class wbr 5100  ^◡ccnv 5646  [cec 8676   ≀ ccoss 38682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-11 2191  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ec 8680  df-coss 39000

This theorem is referenced by:  eqvrelcoss4  39203