Theorem elqsn0m 6839

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 2234	. 2 |- ( A /. R ) = ( A /. R )
2		eleq2 2298	. . 3 |- ( [ y ] R = B -> ( x e. [ y ] R <-> x e. B ) )
3	2	exbidv 1874	. 2 |- ( [ y ] R = B -> ( E. x x e. [ y ] R <-> E. x x e. B ) )
4		eleq2 2298	. . . 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 6813	. . 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 6837	1 |- ( ( dom R = A /\ B e. ( A /. R ) ) -> E. x x e. B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2205   dom cdm 4751   [cec 6767   /.cqs 6768

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-xp 4757  df-cnv 4759  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-ec 6771  df-qs 6775

This theorem is referenced by:  elqsn0  6840  ecelqsdm  6841