Theorem qsinxp 6766

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 6765	. . . . 5 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝑥]𝑅 = [𝑥](𝑅 ∩ (𝐴 × 𝐴)))
2	1	eqeq2d 2241	. . . 4 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝑥](𝑅 ∩ (𝐴 × 𝐴))))
3	2	rexbidva 2527	. . 3 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐴 → (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅 ↔ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝑅 ∩ (𝐴 × 𝐴))))
4	3	abbidv 2347	. 2 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐴 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝑅 ∩ (𝐴 × 𝐴))})
5		df-qs 6694	. 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
6		df-qs 6694	. 2 ⊢ (𝐴 / (𝑅 ∩ (𝐴 × 𝐴))) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝑅 ∩ (𝐴 × 𝐴))}
7	4, 5, 6	3eqtr4g 2287	1 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐴 → (𝐴 / 𝑅) = (𝐴 / (𝑅 ∩ (𝐴 × 𝐴))))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  {cab 2215  ∃wrex 2509   ∩ cin 3196   ⊆ wss 3197   × cxp 4717   “ cima 4722  [cec 6686   / cqs 6687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-ec 6690  df-qs 6694

This theorem is referenced by: (None)