Theorem ssrelrn 4967

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 4966	. . . . 5 |- ( Y e. ran R -> ( Y e. ran R <-> E. a a R Y ) )
2		ssbr 4169	. . . . . . . . . . 11 |- ( R C_ ( A X. B ) -> ( a R Y -> a ( A X. B ) Y ) )
3		brxp 4800	. . . . . . . . . . . 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 1933	. . . . . . 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 2534	. 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 1545   e. wcel 2209   E.wrex 2529   C_ wss 3220   class class class wbr 4125   X. cxp 4767   ran crn 4770

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-cnv 4777  df-dm 4779  df-rn 4780

This theorem is referenced by:  incistruhgr  16245