Theorem rnin 3464

Description: The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60.

Assertion

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

Proof of Theorem rnin

Step	Hyp	Ref	Expression
1		cnvin 3462	. . . 4 |- `'(A i^i B) = (`'A i^i `'B)
2	1	dmeqi 3318	. . 3 |- dom `'(A i^i B) = dom (`'A i^i `'B)
3		dmin 3324	. . 3 |- dom (`'A i^i `'B) (_ (dom `' A i^i dom `' B)
4	2, 3	eqsstr 2094	. 2 |- dom `'(A i^i B) (_ (dom `' A i^i dom `' B)
5		df-rn 3195	. 2 |- ran ( A i^i B) = dom `'(A i^i B)
6		df-rn 3195	. . 3 |- ran A = dom `' A
7		df-rn 3195	. . 3 |- ran B = dom `' B
8	6, 7	ineq12i 2218	. 2 |- (ran A i^i ran B) = (dom `' A i^i dom `' B)
9	4, 5, 8	3sstr4 2103	1 |- ran ( A i^i B) (_ (ran A i^i ran B)

Syntax hints:   i^i cin 2049   (_ wss 2050  `'ccnv 3175  dom cdm 3176  ran crn 3177

This theorem is referenced by:  infxpidmlem11 7563

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-cnv 3192  df-dm 3194  df-rn 3195

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