Theorem ssrelrn 4920

Description: If a relation is a subset of a cartesian product, then for each element of the range of the relation there is an element of the first set of the cartesian product which is related to the element of the range by the relation. (Contributed by AV, 24-Oct-2020.)

Assertion

Ref	Expression
ssrelrn	|- ( ( R C_ ( A X. B ) /\ Y e. ran R ) -> E. a e. A a R Y )

Distinct variable groups:   A, a   B, a   R, a   Y, a

Proof of Theorem ssrelrn

Step	Hyp	Ref	Expression
1		elrng 4919	. . . . 5 |- ( Y e. ran R -> ( Y e. ran R <-> E. a a R Y ) )
2		ssbr 4130	. . . . . . . . . . 11 |- ( R C_ ( A X. B ) -> ( a R Y -> a ( A X. B ) Y ) )
3		brxp 4754	. . . . . . . . . . . 12 |- ( a ( A X. B ) Y <-> ( a e. A /\ Y e. B ) )
4	3	simplbi 274	. . . . . . . . . . 11 |- ( a ( A X. B ) Y -> a e. A )
5	2, 4	syl6 33	. . . . . . . . . 10 |- ( R C_ ( A X. B ) -> ( a R Y -> a e. A ) )
6	5	ancrd 326	. . . . . . . . 9 |- ( R C_ ( A X. B ) -> ( a R Y -> ( a e. A /\ a R Y ) ) )
7	6	adantl 277	. . . . . . . 8 |- ( ( Y e. ran R /\ R C_ ( A X. B ) ) -> ( a R Y -> ( a e. A /\ a R Y ) ) )
8	7	eximdv 1926	. . . . . . 7 |- ( ( Y e. ran R /\ R C_ ( A X. B ) ) -> ( E. a a R Y -> E. a ( a e. A /\ a R Y ) ) )
9	8	ex 115	. . . . . 6 |- ( Y e. ran R -> ( R C_ ( A X. B ) -> ( E. a a R Y -> E. a ( a e. A /\ a R Y ) ) ) )
10	9	com23 78	. . . . 5 |- ( Y e. ran R -> ( E. a a R Y -> ( R C_ ( A X. B ) -> E. a ( a e. A /\ a R Y ) ) ) )
11	1, 10	sylbid 150	. . . 4 |- ( Y e. ran R -> ( Y e. ran R -> ( R C_ ( A X. B ) -> E. a ( a e. A /\ a R Y ) ) ) )
12	11	pm2.43i 49	. . 3 |- ( Y e. ran R -> ( R C_ ( A X. B ) -> E. a ( a e. A /\ a R Y ) ) )
13	12	impcom 125	. 2 |- ( ( R C_ ( A X. B ) /\ Y e. ran R ) -> E. a ( a e. A /\ a R Y ) )
14		df-rex 2514	. 2 |- ( E. a e. A a R Y <-> E. a ( a e. A /\ a R Y ) )
15	13, 14	sylibr 134	1 |- ( ( R C_ ( A X. B ) /\ Y e. ran R ) -> E. a e. A a R Y )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1538   e. wcel 2200   E.wrex 2509   C_ wss 3198   class class class wbr 4086   X. cxp 4721   ran crn 4724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-xp 4729  df-cnv 4731  df-dm 4733  df-rn 4734

This theorem is referenced by:  incistruhgr  15931