Theorem rnin 4676

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 ∩ B) ⊆ (ran A ∩ ran B)

Proof of Theorem rnin

Step	Hyp	Ref	Expression
1		cnvin 4674	. . . 4 ⊢ ◡(A ∩ B) = (◡A ∩ ◡B)
2	1	dmeqi 4479	. . 3 ⊢ dom ◡(A ∩ B) = dom (◡A ∩ ◡B)
3		dmin 4486	. . 3 ⊢ dom (◡A ∩ ◡B) ⊆ (dom ◡A ∩ dom ◡B)
4	2, 3	eqsstri 2969	. 2 ⊢ dom ◡(A ∩ B) ⊆ (dom ◡A ∩ dom ◡B)
5		df-rn 4299	. 2 ⊢ ran (A ∩ B) = dom ◡(A ∩ B)
6		df-rn 4299	. . 3 ⊢ ran A = dom ◡A
7		df-rn 4299	. . 3 ⊢ ran B = dom ◡B
8	6, 7	ineq12i 3130	. 2 ⊢ (ran A ∩ ran B) = (dom ◡A ∩ dom ◡B)
9	4, 5, 8	3sstr4i 2978	1 ⊢ ran (A ∩ B) ⊆ (ran A ∩ ran B)

Syntax hints:   ∩ cin 2910   ⊆ wss 2911  ^◡ccnv 4287  dom cdm 4288  ran crn 4289

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299

This theorem is referenced by:  inimass  4683