Theorem ecinxp 8832

Description: Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.)

Assertion

Ref	Expression
ecinxp	⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 = [𝐵](𝑅 ∩ (𝐴 × 𝐴)))

Proof of Theorem ecinxp

Step	Hyp	Ref	Expression
1		simpr 484	. . . . . . . 8 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴)
2	1	snssd 4809	. . . . . . 7 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝐵} ⊆ 𝐴)
3		dfss2 3969	. . . . . . 7 ⊢ ({𝐵} ⊆ 𝐴 ↔ ({𝐵} ∩ 𝐴) = {𝐵})
4	2, 3	sylib 218	. . . . . 6 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → ({𝐵} ∩ 𝐴) = {𝐵})
5	4	imaeq2d 6078	. . . . 5 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑅 “ ({𝐵} ∩ 𝐴)) = (𝑅 “ {𝐵}))
6	5	ineq1d 4219	. . . 4 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝑅 “ ({𝐵} ∩ 𝐴)) ∩ 𝐴) = ((𝑅 “ {𝐵}) ∩ 𝐴))
7		imass2 6120	. . . . . . 7 ⊢ ({𝐵} ⊆ 𝐴 → (𝑅 “ {𝐵}) ⊆ (𝑅 “ 𝐴))
8	2, 7	syl 17	. . . . . 6 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑅 “ {𝐵}) ⊆ (𝑅 “ 𝐴))
9		simpl 482	. . . . . 6 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑅 “ 𝐴) ⊆ 𝐴)
10	8, 9	sstrd 3994	. . . . 5 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑅 “ {𝐵}) ⊆ 𝐴)
11		dfss2 3969	. . . . 5 ⊢ ((𝑅 “ {𝐵}) ⊆ 𝐴 ↔ ((𝑅 “ {𝐵}) ∩ 𝐴) = (𝑅 “ {𝐵}))
12	10, 11	sylib 218	. . . 4 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝑅 “ {𝐵}) ∩ 𝐴) = (𝑅 “ {𝐵}))
13	6, 12	eqtr2d 2778	. . 3 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑅 “ {𝐵}) = ((𝑅 “ ({𝐵} ∩ 𝐴)) ∩ 𝐴))
14		imainrect 6201	. . 3 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) “ {𝐵}) = ((𝑅 “ ({𝐵} ∩ 𝐴)) ∩ 𝐴)
15	13, 14	eqtr4di 2795	. 2 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑅 “ {𝐵}) = ((𝑅 ∩ (𝐴 × 𝐴)) “ {𝐵}))
16		df-ec 8747	. 2 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵})
17		df-ec 8747	. 2 ⊢ [𝐵](𝑅 ∩ (𝐴 × 𝐴)) = ((𝑅 ∩ (𝐴 × 𝐴)) “ {𝐵})
18	15, 16, 17	3eqtr4g 2802	1 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 = [𝐵](𝑅 ∩ (𝐴 × 𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ∩ cin 3950   ⊆ wss 3951  {csn 4626   × cxp 5683   “ cima 5688  [cec 8743

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 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747

This theorem is referenced by:  qsinxp  8833  qusin  17589  pi1addval  25081  pi1grplem  25082