Theorem rnin 6145

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 5406 and ax-sep 5259. (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 4189	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2	1	rnssi 5932	. 2 ⊢ ran (𝐴 ∩ 𝐵) ⊆ ran 𝐴
3		inss2 4190	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
4	3	rnssi 5932	. 2 ⊢ ran (𝐴 ∩ 𝐵) ⊆ ran 𝐵
5	2, 4	ssini 4192	1 ⊢ ran (𝐴 ∩ 𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵)

This proof depends on syntax axioms:   ∩ cin 3905   ⊆ wss 3906  ran crn 5664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-cnv 5671  df-dm 5673  df-rn 5674

This theorem is used by:  inimass  6154  restutop  24445