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Theorem qsel 6605
Description: If an element of a quotient set contains a given element, it is equal to the equivalence class of the element. (Contributed by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
qsel ((𝑅 Er 𝑋𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶𝐵) → 𝐵 = [𝐶]𝑅)

Proof of Theorem qsel
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2177 . . 3 (𝐴 / 𝑅) = (𝐴 / 𝑅)
2 eleq2 2241 . . . 4 ([𝑥]𝑅 = 𝐵 → (𝐶 ∈ [𝑥]𝑅𝐶𝐵))
3 eqeq1 2184 . . . 4 ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 = [𝐶]𝑅𝐵 = [𝐶]𝑅))
42, 3imbi12d 234 . . 3 ([𝑥]𝑅 = 𝐵 → ((𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅) ↔ (𝐶𝐵𝐵 = [𝐶]𝑅)))
5 vex 2740 . . . . . 6 𝑥 ∈ V
6 elecg 6566 . . . . . 6 ((𝐶 ∈ [𝑥]𝑅𝑥 ∈ V) → (𝐶 ∈ [𝑥]𝑅𝑥𝑅𝐶))
75, 6mpan2 425 . . . . 5 (𝐶 ∈ [𝑥]𝑅 → (𝐶 ∈ [𝑥]𝑅𝑥𝑅𝐶))
87ibi 176 . . . 4 (𝐶 ∈ [𝑥]𝑅𝑥𝑅𝐶)
9 simpll 527 . . . . . 6 (((𝑅 Er 𝑋𝑥𝐴) ∧ 𝑥𝑅𝐶) → 𝑅 Er 𝑋)
10 simpr 110 . . . . . 6 (((𝑅 Er 𝑋𝑥𝐴) ∧ 𝑥𝑅𝐶) → 𝑥𝑅𝐶)
119, 10erthi 6574 . . . . 5 (((𝑅 Er 𝑋𝑥𝐴) ∧ 𝑥𝑅𝐶) → [𝑥]𝑅 = [𝐶]𝑅)
1211ex 115 . . . 4 ((𝑅 Er 𝑋𝑥𝐴) → (𝑥𝑅𝐶 → [𝑥]𝑅 = [𝐶]𝑅))
138, 12syl5 32 . . 3 ((𝑅 Er 𝑋𝑥𝐴) → (𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅))
141, 4, 13ectocld 6594 . 2 ((𝑅 Er 𝑋𝐵 ∈ (𝐴 / 𝑅)) → (𝐶𝐵𝐵 = [𝐶]𝑅))
15143impia 1200 1 ((𝑅 Er 𝑋𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶𝐵) → 𝐵 = [𝐶]𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wcel 2148  Vcvv 2737   class class class wbr 4000   Er wer 6525  [cec 6526   / cqs 6527
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-er 6528  df-ec 6530  df-qs 6534
This theorem is referenced by: (None)
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