Theorem alrmomo 34123

Description: Lemma for ineccnvmo2 34125. (Contributed by Peter Mazsa, 3-Sep-2021.)

Assertion

Ref	Expression
alrmomo	|- ( A. x E* y e. ran R x R y <-> A. x E* y x R y )

Proof of Theorem alrmomo

Step	Hyp	Ref	Expression
1		df-rmo 2920	. . 3 |- ( E* y e. ran R x R y <-> E* y ( y e. ran R /\ x R y ) )
2		cnvresrn 34116	. . . . . 6 |- ( `' R |` ran R ) = `' R
3	2	breqi 4659	. . . . 5 |- ( y ( `' R |` ran R ) x <-> y `' R x )
4		brresALTV 34032	. . . . . . 7 |- ( x e. _V -> ( y ( `' R |` ran R ) x <-> ( y e. ran R /\ y `' R x ) ) )
5	4	elv 33983	. . . . . 6 |- ( y ( `' R |` ran R ) x <-> ( y e. ran R /\ y `' R x ) )
6		brcnvg 5303	. . . . . . . 8 |- ( ( y e. _V /\ x e. _V ) -> ( y `' R x <-> x R y ) )
7	6	el2v 33984	. . . . . . 7 |- ( y `' R x <-> x R y )
8	7	anbi2i 730	. . . . . 6 |- ( ( y e. ran R /\ y `' R x ) <-> ( y e. ran R /\ x R y ) )
9	5, 8	bitri 264	. . . . 5 |- ( y ( `' R |` ran R ) x <-> ( y e. ran R /\ x R y ) )
10	3, 9, 7	3bitr3i 290	. . . 4 |- ( ( y e. ran R /\ x R y ) <-> x R y )
11	10	mobii 2493	. . 3 |- ( E* y ( y e. ran R /\ x R y ) <-> E* y x R y )
12	1, 11	bitri 264	. 2 |- ( E* y e. ran R x R y <-> E* y x R y )
13	12	albii 1747	1 |- ( A. x E* y e. ran R x R y <-> A. x E* y x R y )

Syntax hints:   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   E*wmo 2471   E*wrmo 2915   _Vcvv 3200   class class class wbr 4653   `'ccnv 5113   ran crn 5115   |` cres 5116

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-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by:  ineccnvmo2  34125