Theorem rnin 5542

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 5540	. . . 4 |- `' ( A i^i B ) = ( `' A i^i `' B )
2	1	dmeqi 5325	. . 3 |- dom `' ( A i^i B ) = dom ( `' A i^i `' B )
3		dmin 5332	. . 3 |- dom ( `' A i^i `' B ) C_ ( dom `' A i^i dom `' B )
4	2, 3	eqsstri 3635	. 2 |- dom `' ( A i^i B ) C_ ( dom `' A i^i dom `' B )
5		df-rn 5125	. 2 |- ran ( A i^i B ) = dom `' ( A i^i B )
6		df-rn 5125	. . 3 |- ran A = dom `' A
7		df-rn 5125	. . 3 |- ran B = dom `' B
8	6, 7	ineq12i 3812	. 2 |- ( ran A i^i ran B ) = ( dom `' A i^i dom `' B )
9	4, 5, 8	3sstr4i 3644	1 |- ran ( A i^i B ) C_ ( ran A i^i ran B )

Syntax hints:   i^i cin 3573   C_ wss 3574   `'ccnv 5113   dom cdm 5114   ran crn 5115

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-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

This theorem is referenced by:  inimass  5549  restutop  22041