Theorem rnin 5075

Description: The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)

Assertion

Ref	Expression
rnin	|- ran ( A i^i B ) C_ ( ran A i^i ran B )

Proof of Theorem rnin

Step	Hyp	Ref	Expression
1		cnvin 5073	. . . 4 |- `' ( A i^i B ) = ( `' A i^i `' B )
2	1	dmeqi 4863	. . 3 |- dom `' ( A i^i B ) = dom ( `' A i^i `' B )
3		dmin 4870	. . 3 |- dom ( `' A i^i `' B ) C_ ( dom `' A i^i dom `' B )
4	2, 3	eqsstri 3211	. 2 |- dom `' ( A i^i B ) C_ ( dom `' A i^i dom `' B )
5		df-rn 4670	. 2 |- ran ( A i^i B ) = dom `' ( A i^i B )
6		df-rn 4670	. . 3 |- ran A = dom `' A
7		df-rn 4670	. . 3 |- ran B = dom `' B
8	6, 7	ineq12i 3358	. 2 |- ( ran A i^i ran B ) = ( dom `' A i^i dom `' B )
9	4, 5, 8	3sstr4i 3220	1 |- ran ( A i^i B ) C_ ( ran A i^i ran B )

Syntax hints:   i^i cin 3152   C_ wss 3153   `'ccnv 4658   dom cdm 4659   ran crn 4660

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-dm 4669  df-rn 4670

This theorem is referenced by:  inimass  5082