Theorem qsinxp 8851

Description: Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)

Assertion

Ref	Expression
qsinxp	⊢ ((𝑅 “ 𝐴) ⊆ 𝐴 → (𝐴 / 𝑅) = (𝐴 / (𝑅 ∩ (𝐴 × 𝐴))))

Proof of Theorem qsinxp

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ecinxp 8850	. . . . 5 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝑥]𝑅 = [𝑥](𝑅 ∩ (𝐴 × 𝐴)))
2	1	eqeq2d 2751	. . . 4 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝑥](𝑅 ∩ (𝐴 × 𝐴))))
3	2	rexbidva 3183	. . 3 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐴 → (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅 ↔ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝑅 ∩ (𝐴 × 𝐴))))
4	3	abbidv 2811	. 2 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐴 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝑅 ∩ (𝐴 × 𝐴))})
5		df-qs 8769	. 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
6		df-qs 8769	. 2 ⊢ (𝐴 / (𝑅 ∩ (𝐴 × 𝐴))) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝑅 ∩ (𝐴 × 𝐴))}
7	4, 5, 6	3eqtr4g 2805	1 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐴 → (𝐴 / 𝑅) = (𝐴 / (𝑅 ∩ (𝐴 × 𝐴))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∃wrex 3076   ∩ cin 3975   ⊆ wss 3976   × cxp 5698   “ cima 5703  [cec 8761   / cqs 8762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ec 8765  df-qs 8769

This theorem is referenced by:  pi1buni  25092  pi1bas3  25095