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Theorem qsel 7944
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 2724 . . 3 (𝐴 / 𝑅) = (𝐴 / 𝑅)
2 eleq2 2792 . . . 4 ([𝑥]𝑅 = 𝐵 → (𝐶 ∈ [𝑥]𝑅𝐶𝐵))
3 eqeq1 2728 . . . 4 ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 = [𝐶]𝑅𝐵 = [𝐶]𝑅))
42, 3imbi12d 333 . . 3 ([𝑥]𝑅 = 𝐵 → ((𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅) ↔ (𝐶𝐵𝐵 = [𝐶]𝑅)))
5 vex 3307 . . . . . 6 𝑥 ∈ V
6 elecg 7903 . . . . . 6 ((𝐶 ∈ [𝑥]𝑅𝑥 ∈ V) → (𝐶 ∈ [𝑥]𝑅𝑥𝑅𝐶))
75, 6mpan2 709 . . . . 5 (𝐶 ∈ [𝑥]𝑅 → (𝐶 ∈ [𝑥]𝑅𝑥𝑅𝐶))
87ibi 256 . . . 4 (𝐶 ∈ [𝑥]𝑅𝑥𝑅𝐶)
9 simpll 807 . . . . . 6 (((𝑅 Er 𝑋𝑥𝐴) ∧ 𝑥𝑅𝐶) → 𝑅 Er 𝑋)
10 simpr 479 . . . . . 6 (((𝑅 Er 𝑋𝑥𝐴) ∧ 𝑥𝑅𝐶) → 𝑥𝑅𝐶)
119, 10erthi 7911 . . . . 5 (((𝑅 Er 𝑋𝑥𝐴) ∧ 𝑥𝑅𝐶) → [𝑥]𝑅 = [𝐶]𝑅)
1211ex 449 . . . 4 ((𝑅 Er 𝑋𝑥𝐴) → (𝑥𝑅𝐶 → [𝑥]𝑅 = [𝐶]𝑅))
138, 12syl5 34 . . 3 ((𝑅 Er 𝑋𝑥𝐴) → (𝐶 ∈ [𝑥]𝑅 → [𝑥]𝑅 = [𝐶]𝑅))
141, 4, 13ectocld 7932 . 2 ((𝑅 Er 𝑋𝐵 ∈ (𝐴 / 𝑅)) → (𝐶𝐵𝐵 = [𝐶]𝑅))
15143impia 1109 1 ((𝑅 Er 𝑋𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶𝐵) → 𝐵 = [𝐶]𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1596  wcel 2103  Vcvv 3304   class class class wbr 4760   Er wer 7859  [cec 7860   / cqs 7861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-er 7862  df-ec 7864  df-qs 7868
This theorem is referenced by:  frgpnabllem2  18398  prter3  34588
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