Theorem dmecd 34074

Description: Equality of the coset of B and the coset of C implies equivalence of domain elementhood (equivalence is not necessary as opposed to ereldm 7790). (Contributed by Peter Mazsa, 9-Oct-2018.)

Hypotheses

Ref	Expression
dmecd.1	|- ( ph -> dom R = A )
dmecd.2	|- ( ph -> [ B ] R = [ C ] R )

Assertion

Ref	Expression
dmecd	|- ( ph -> ( B e. A <-> C e. A ) )

Proof of Theorem dmecd

Step	Hyp	Ref	Expression
1		dmecd.2	. . . 4 |- ( ph -> [ B ] R = [ C ] R )
2	1	neeq1d 2853	. . 3 |- ( ph -> ( [ B ] R =/= (/) <-> [ C ] R =/= (/) ) )
3		ecdmn0 7789	. . 3 |- ( B e. dom R <-> [ B ] R =/= (/) )
4		ecdmn0 7789	. . 3 |- ( C e. dom R <-> [ C ] R =/= (/) )
5	2, 3, 4	3bitr4g 303	. 2 |- ( ph -> ( B e. dom R <-> C e. dom R ) )
6		dmecd.1	. . 3 |- ( ph -> dom R = A )
7	6	eleq2d 2687	. 2 |- ( ph -> ( B e. dom R <-> B e. A ) )
8	6	eleq2d 2687	. 2 |- ( ph -> ( C e. dom R <-> C e. A ) )
9	5, 7, 8	3bitr3d 298	1 |- ( ph -> ( B e. A <-> C e. A ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   dom cdm 5114   [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-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744

This theorem is referenced by:  dmec2d  34075