Theorem elqsn0m 6569

Description: An element of a quotient set is inhabited. (Contributed by Jim Kingdon, 21-Aug-2019.)

Assertion

Ref	Expression
elqsn0m	|- ( ( dom R = A /\ B e. ( A /. R ) ) -> E. x x e. B )

Distinct variable groups:   x, R   x, A   x, B

Proof of Theorem elqsn0m

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2165	. 2 |- ( A /. R ) = ( A /. R )
2		eleq2 2230	. . 3 |- ( [ y ] R = B -> ( x e. [ y ] R <-> x e. B ) )
3	2	exbidv 1813	. 2 |- ( [ y ] R = B -> ( E. x x e. [ y ] R <-> E. x x e. B ) )
4		eleq2 2230	. . . 4 |- ( dom R = A -> ( y e. dom R <-> y e. A ) )
5	4	biimpar 295	. . 3 |- ( ( dom R = A /\ y e. A ) -> y e. dom R )
6		ecdmn0m 6543	. . 3 |- ( y e. dom R <-> E. x x e. [ y ] R )
7	5, 6	sylib 121	. 2 |- ( ( dom R = A /\ y e. A ) -> E. x x e. [ y ] R )
8	1, 3, 7	ectocld 6567	1 |- ( ( dom R = A /\ B e. ( A /. R ) ) -> E. x x e. B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   E.wex 1480   e. wcel 2136   dom cdm 4604   [cec 6499   /.cqs 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-ec 6503  df-qs 6507

This theorem is referenced by:  elqsn0  6570  ecelqsdm  6571