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Theorem qsinxp 6081
Description: Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
Assertion
Ref Expression
qsinxp ((𝑅A) ⊆ A → (A / 𝑅) = (A / (𝑅 ∩ (A × A))))

Proof of Theorem qsinxp
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecinxp 6080 . . . . 5 (((𝑅A) ⊆ A x A) → [x]𝑅 = [x](𝑅 ∩ (A × A)))
21eqeq2d 2024 . . . 4 (((𝑅A) ⊆ A x A) → (y = [x]𝑅y = [x](𝑅 ∩ (A × A))))
32rexbidva 2292 . . 3 ((𝑅A) ⊆ A → (x A y = [x]𝑅x A y = [x](𝑅 ∩ (A × A))))
43abbidv 2128 . 2 ((𝑅A) ⊆ A → {yx A y = [x]𝑅} = {yx A y = [x](𝑅 ∩ (A × A))})
5 df-qs 6011 . 2 (A / 𝑅) = {yx A y = [x]𝑅}
6 df-qs 6011 . 2 (A / (𝑅 ∩ (A × A))) = {yx A y = [x](𝑅 ∩ (A × A))}
74, 5, 63eqtr4g 2070 1 ((𝑅A) ⊆ A → (A / 𝑅) = (A / (𝑅 ∩ (A × A))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1223   wcel 1366  {cab 1999  wrex 2276  cin 2884  wss 2885   × cxp 4258  cima 4263  [cec 6003   / cqs 6004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-pow 3890  ax-pr 3907
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-br 3728  df-opab 3782  df-xp 4266  df-rel 4267  df-cnv 4268  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-ec 6007  df-qs 6011
This theorem is referenced by: (None)
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