Theorem rnin 6130

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 5390 and ax-sep 5246. (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 4188	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2	1	rnssi 5916	. 2 ⊢ ran (𝐴 ∩ 𝐵) ⊆ ran 𝐴
3		inss2 4189	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
4	3	rnssi 5916	. 2 ⊢ ran (𝐴 ∩ 𝐵) ⊆ ran 𝐵
5	2, 4	ssini 4191	1 ⊢ ran (𝐴 ∩ 𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵)

Syntax hints:   ∩ cin 3903   ⊆ wss 3904  ran crn 5648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-cnv 5655  df-dm 5657  df-rn 5658

This theorem is referenced by:  inimass  6140  restutop  24294