Theorem elqsn0m 6427

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 2100	. 2 |- ( A /. R ) = ( A /. R )
2		eleq2 2163	. . 3 |- ( [ y ] R = B -> ( x e. [ y ] R <-> x e. B ) )
3	2	exbidv 1764	. 2 |- ( [ y ] R = B -> ( E. x x e. [ y ] R <-> E. x x e. B ) )
4		eleq2 2163	. . . 4 |- ( dom R = A -> ( y e. dom R <-> y e. A ) )
5	4	biimpar 293	. . 3 |- ( ( dom R = A /\ y e. A ) -> y e. dom R )
6		ecdmn0m 6401	. . 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 6425	1 |- ( ( dom R = A /\ B e. ( A /. R ) ) -> E. x x e. B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1299   E.wex 1436   e. wcel 1448   dom cdm 4477   [cec 6357   /.cqs 6358

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-xp 4483  df-cnv 4485  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-ec 6361  df-qs 6365

This theorem is referenced by:  elqsn0  6428  ecelqsdm  6429