Theorem alrmomo2 34124

Description: Lemma for inecmo3 34126. (Contributed by Peter Mazsa, 5-Sep-2021.)

Assertion

Ref	Expression
alrmomo2	|- ( Rel R -> ( A. x E* u e. dom R u R x <-> A. x E* u u R x ) )

Distinct variable groups:   u, R   x, R

Proof of Theorem alrmomo2

Step	Hyp	Ref	Expression
1		df-rmo 2920	. . 3 |- ( E* u e. dom R u R x <-> E* u ( u e. dom R /\ u R x ) )
2		brresALTV 34032	. . . . . 6 |- ( x e. _V -> ( u ( R |` dom R ) x <-> ( u e. dom R /\ u R x ) ) )
3	2	elv 33983	. . . . 5 |- ( u ( R |` dom R ) x <-> ( u e. dom R /\ u R x ) )
4		resdm 5441	. . . . . 6 |- ( Rel R -> ( R |` dom R ) = R )
5	4	breqd 4664	. . . . 5 |- ( Rel R -> ( u ( R |` dom R ) x <-> u R x ) )
6	3, 5	syl5bbr 274	. . . 4 |- ( Rel R -> ( ( u e. dom R /\ u R x ) <-> u R x ) )
7	6	mobidv 2491	. . 3 |- ( Rel R -> ( E* u ( u e. dom R /\ u R x ) <-> E* u u R x ) )
8	1, 7	syl5bb 272	. 2 |- ( Rel R -> ( E* u e. dom R u R x <-> E* u u R x ) )
9	8	albidv 1849	1 |- ( Rel R -> ( A. x E* u e. dom R u R x <-> A. x E* u u R x ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   E*wmo 2471   E*wrmo 2915   _Vcvv 3200   class class class wbr 4653   dom cdm 5114   |` cres 5116   Rel wrel 5119

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-ral 2917  df-rex 2918  df-rmo 2920  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-rel 5121  df-dm 5124  df-res 5126

This theorem is referenced by:  inecmo3  34126