Theorem eccnvepres 34045

Description: Restricted converse epsilon coset of B. (Contributed by Peter Mazsa, 11-Feb-2018.) (Revised by Peter Mazsa, 21-Oct-2021.)

Assertion

Ref	Expression
eccnvepres	|- ( B e. V -> [ B ] ( `' _E |` A ) = { x e. B | B e. A } )

Distinct variable groups:   x, A   x, B   x, V

Proof of Theorem eccnvepres

Step	Hyp	Ref	Expression
1		brcnvep 34029	. . . 4 |- ( B e. V -> ( B `' _E x <-> x e. B ) )
2	1	anbi1cd 33997	. . 3 |- ( B e. V -> ( ( B e. A /\ B `' _E x ) <-> ( x e. B /\ B e. A ) ) )
3	2	abbidv 2741	. 2 |- ( B e. V -> { x | ( B e. A /\ B `' _E x ) } = { x | ( x e. B /\ B e. A ) } )
4		ecres 34043	. 2 |- [ B ] ( `' _E |` A ) = { x | ( B e. A /\ B `' _E x ) }
5		df-rab 2921	. 2 |- { x e. B | B e. A } = { x | ( x e. B /\ B e. A ) }
6	3, 4, 5	3eqtr4g 2681	1 |- ( B e. V -> [ B ] ( `' _E |` A ) = { x e. B | B e. A } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   class class class wbr 4653   _E cep 5028   `'ccnv 5113   |` cres 5116   [cec 7740

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-eprel 5029  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744

This theorem is referenced by: (None)