Theorem elqsn0m 6581

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 2170	. 2 |- ( A /. R ) = ( A /. R )
2		eleq2 2234	. . 3 |- ( [ y ] R = B -> ( x e. [ y ] R <-> x e. B ) )
3	2	exbidv 1818	. 2 |- ( [ y ] R = B -> ( E. x x e. [ y ] R <-> E. x x e. B ) )
4		eleq2 2234	. . . 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 6555	. . 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 6579	1 |- ( ( dom R = A /\ B e. ( A /. R ) ) -> E. x x e. B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   dom cdm 4611   [cec 6511   /.cqs 6512

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-ec 6515  df-qs 6519

This theorem is referenced by:  elqsn0  6582  ecelqsdm  6583