Theorem prtlem11 34151

Description: Lemma for prter2 34166. (Contributed by Rodolfo Medina, 12-Oct-2010.)

Assertion

Ref	Expression
prtlem11	|- ( B e. D -> ( C e. A -> ( B = [ C ] .~ -> B e. ( A /. .~ ) ) ) )

Proof of Theorem prtlem11

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		risset 3062	. . . 4 |- ( C e. A <-> E. x e. A x = C )
2		r19.41v 3089	. . . . 5 |- ( E. x e. A ( x = C /\ B = [ C ] .~ ) <-> ( E. x e. A x = C /\ B = [ C ] .~ ) )
3		eceq1 7782	. . . . . . 7 |- ( x = C -> [ x ] .~ = [ C ] .~ )
4		eqtr3 2643	. . . . . . . 8 |- ( ( [ x ] .~ = [ C ] .~ /\ B = [ C ] .~ ) -> [ x ] .~ = B )
5	4	eqcomd 2628	. . . . . . 7 |- ( ( [ x ] .~ = [ C ] .~ /\ B = [ C ] .~ ) -> B = [ x ] .~ )
6	3, 5	sylan 488	. . . . . 6 |- ( ( x = C /\ B = [ C ] .~ ) -> B = [ x ] .~ )
7	6	reximi 3011	. . . . 5 |- ( E. x e. A ( x = C /\ B = [ C ] .~ ) -> E. x e. A B = [ x ] .~ )
8	2, 7	sylbir 225	. . . 4 |- ( ( E. x e. A x = C /\ B = [ C ] .~ ) -> E. x e. A B = [ x ] .~ )
9	1, 8	sylanb 489	. . 3 |- ( ( C e. A /\ B = [ C ] .~ ) -> E. x e. A B = [ x ] .~ )
10		elqsg 7798	. . 3 |- ( B e. D -> ( B e. ( A /. .~ ) <-> E. x e. A B = [ x ] .~ ) )
11	9, 10	syl5ibr 236	. 2 |- ( B e. D -> ( ( C e. A /\ B = [ C ] .~ ) -> B e. ( A /. .~ ) ) )
12	11	expd 452	1 |- ( B e. D -> ( C e. A -> ( B = [ C ] .~ -> B e. ( A /. .~ ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   [cec 7740   /.cqs 7741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744  df-qs 7748

This theorem is referenced by:  prter2  34166