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Theorem elqsn0m 6085
Description: An element of a quotient set is inhabited. (Contributed by Jim Kingdon, 21-Aug-2019.)
Assertion
Ref Expression
elqsn0m ((dom 𝑅 = A B (A / 𝑅)) → x x B)
Distinct variable groups:   x,𝑅   x,A   x,B

Proof of Theorem elqsn0m
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eqid 2022 . 2 (A / 𝑅) = (A / 𝑅)
2 eleq2 2083 . . 3 ([y]𝑅 = B → (x [y]𝑅x B))
32exbidv 1688 . 2 ([y]𝑅 = B → (x x [y]𝑅x x B))
4 eleq2 2083 . . . 4 (dom 𝑅 = A → (y dom 𝑅y A))
54biimpar 281 . . 3 ((dom 𝑅 = A y A) → y dom 𝑅)
6 ecdmn0m 6059 . . 3 (y dom 𝑅x x [y]𝑅)
75, 6sylib 127 . 2 ((dom 𝑅 = A y A) → x x [y]𝑅)
81, 3, 7ectocld 6083 1 ((dom 𝑅 = A B (A / 𝑅)) → x x B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228  wex 1362   wcel 1374  dom cdm 4272  [cec 6015   / cqs 6016
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-cnv 4280  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-ec 6019  df-qs 6023
This theorem is referenced by:  elqsn0  6086  ecelqsdm  6087
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