Theorem rnin 6143

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 5404 and ax-sep 5257. (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 5930	. 2 ⊢ ran (𝐴 ∩ 𝐵) ⊆ ran 𝐴
3		inss2 4190	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
4	3	rnssi 5930	. 2 ⊢ ran (𝐴 ∩ 𝐵) ⊆ ran 𝐵
5	2, 4	ssini 4192	1 ⊢ ran (𝐴 ∩ 𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵)

Syntax hints:   ∩ cin 3904   ⊆ wss 3905  ran crn 5662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-cnv 5669  df-dm 5671  df-rn 5672

This theorem is referenced by:  inimass  6152  restutop  24394