Theorem elqsn0m 6671

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 2196	. 2 |- ( A /. R ) = ( A /. R )
2		eleq2 2260	. . 3 |- ( [ y ] R = B -> ( x e. [ y ] R <-> x e. B ) )
3	2	exbidv 1839	. 2 |- ( [ y ] R = B -> ( E. x x e. [ y ] R <-> E. x x e. B ) )
4		eleq2 2260	. . . 4 |- ( dom R = A -> ( y e. dom R <-> y e. A ) )
5	4	biimpar 297	. . 3 |- ( ( dom R = A /\ y e. A ) -> y e. dom R )
6		ecdmn0m 6645	. . 3 |- ( y e. dom R <-> E. x x e. [ y ] R )
7	5, 6	sylib 122	. 2 |- ( ( dom R = A /\ y e. A ) -> E. x x e. [ y ] R )
8	1, 3, 7	ectocld 6669	1 |- ( ( dom R = A /\ B e. ( A /. R ) ) -> E. x x e. B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   E.wex 1506   e. wcel 2167   dom cdm 4664   [cec 6599   /.cqs 6600

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-xp 4670  df-cnv 4672  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-ec 6603  df-qs 6607

This theorem is referenced by:  elqsn0  6672  ecelqsdm  6673