Theorem qsinxp 7823

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 7822	. . . . 5 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝑥]𝑅 = [𝑥](𝑅 ∩ (𝐴 × 𝐴)))
2	1	eqeq2d 2632	. . . 4 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝑥](𝑅 ∩ (𝐴 × 𝐴))))
3	2	rexbidva 3049	. . 3 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐴 → (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅 ↔ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝑅 ∩ (𝐴 × 𝐴))))
4	3	abbidv 2741	. 2 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐴 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝑅 ∩ (𝐴 × 𝐴))})
5		df-qs 7748	. 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
6		df-qs 7748	. 2 ⊢ (𝐴 / (𝑅 ∩ (𝐴 × 𝐴))) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝑅 ∩ (𝐴 × 𝐴))}
7	4, 5, 6	3eqtr4g 2681	1 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐴 → (𝐴 / 𝑅) = (𝐴 / (𝑅 ∩ (𝐴 × 𝐴))))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  ∃wrex 2913   ∩ cin 3573   ⊆ wss 3574   × cxp 5112   “ cima 5117  [cec 7740   / cqs 7741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744  df-qs 7748

This theorem is referenced by:  pi1buni  22840  pi1bas3  22843