Theorem rnin 6144

Description: The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.) Avoid ax-pr 5405 and ax-sep 5261. (Revised by Umit Teoman Dogan, 10-Jun-2026.)

Assertion

Ref	Expression
rnin	⊢ ran (𝐴 ∩ 𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵)

Proof of Theorem rnin

Step	Hyp	Ref	Expression
1		inss1 4197	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2	1	rnssi 5931	. 2 ⊢ ran (𝐴 ∩ 𝐵) ⊆ ran 𝐴
3		inss2 4198	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
4	3	rnssi 5931	. 2 ⊢ ran (𝐴 ∩ 𝐵) ⊆ ran 𝐵
5	2, 4	ssini 4200	1 ⊢ ran (𝐴 ∩ 𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵)

Syntax hints:   ∩ cin 3912   ⊆ wss 3913  ran crn 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-cnv 5670  df-dm 5672  df-rn 5673

This theorem is referenced by:  inimass  6153  restutop  24362