Theorem ssrelrn 4947

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 4946	. . . . 5 |- ( Y e. ran R -> ( Y e. ran R <-> E. a a R Y ) )
2		ssbr 4153	. . . . . . . . . . 11 |- ( R C_ ( A X. B ) -> ( a R Y -> a ( A X. B ) Y ) )
3		brxp 4780	. . . . . . . . . . . 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 1929	. . . . . . 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 2526	. 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 1541   e. wcel 2203   E.wrex 2521   C_ wss 3211   class class class wbr 4109   X. cxp 4747   ran crn 4750

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-cnv 4757  df-dm 4759  df-rn 4760

This theorem is referenced by:  incistruhgr  16085