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Theorem uniqs 5984
 Description: The union of a quotient set. (Contributed by set.mm contributors, 9-Dec-2008.)
Assertion
Ref Expression
uniqs (R V(A / R) = (RA))

Proof of Theorem uniqs
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecexg 5949 . . . . 5 (R V → [x]R V)
21ralrimivw 2698 . . . 4 (R Vx A [x]R V)
3 dfiun2g 3999 . . . 4 (x A [x]R V → x A [x]R = {y x A y = [x]R})
42, 3syl 15 . . 3 (R Vx A [x]R = {y x A y = [x]R})
54eqcomd 2358 . 2 (R V{y x A y = [x]R} = x A [x]R)
6 df-qs 5951 . . 3 (A / R) = {y x A y = [x]R}
76unieqi 3901 . 2 (A / R) = {y x A y = [x]R}
8 df-ec 5947 . . . . 5 [x]R = (R “ {x})
98a1i 10 . . . 4 (x A → [x]R = (R “ {x}))
109iuneq2i 3987 . . 3 x A [x]R = x A (R “ {x})
11 imaiun 5464 . . 3 (Rx A {x}) = x A (R “ {x})
12 iunid 4021 . . . 4 x A {x} = A
1312imaeq2i 4940 . . 3 (Rx A {x}) = (RA)
1410, 11, 133eqtr2ri 2380 . 2 (RA) = x A [x]R
155, 7, 143eqtr4g 2410 1 (R V(A / R) = (RA))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  ∃wrex 2615  Vcvv 2859  {csn 3737  ∪cuni 3891  ∪ciun 3969   “ cima 4722  [cec 5945   / cqs 5946 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-br 4640  df-ima 4727  df-ec 5947  df-qs 5951 This theorem is referenced by:  uniqs2  5985  ecqs  5988
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