Theorem rnin 6139

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 5391 and ax-sep 5248. (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 4132	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2	1	rnssi 5923	. 2 ⊢ ran (𝐴 ∩ 𝐵) ⊆ ran 𝐴
3		inss2 4133	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
4	3	rnssi 5923	. 2 ⊢ ran (𝐴 ∩ 𝐵) ⊆ ran 𝐵
5	2, 4	ssini 4135	1 ⊢ ran (𝐴 ∩ 𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵)

This proof depends on syntax axioms:   ⊆ wss 3896   ∩ cin 4088  ran crn 5652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-cnv 5659  df-dm 5661  df-rn 5662

This theorem is used by:  inimass  6147  restutop  24565