Theorem rnin 6169

Description: The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)

Assertion

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

Proof of Theorem rnin

Step	Hyp	Ref	Expression
1		cnvin 6167	. . . 4 ⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵)
2	1	dmeqi 5918	. . 3 ⊢ dom ◡(𝐴 ∩ 𝐵) = dom (◡𝐴 ∩ ◡𝐵)
3		dmin 5925	. . 3 ⊢ dom (◡𝐴 ∩ ◡𝐵) ⊆ (dom ◡𝐴 ∩ dom ◡𝐵)
4	2, 3	eqsstri 4030	. 2 ⊢ dom ◡(𝐴 ∩ 𝐵) ⊆ (dom ◡𝐴 ∩ dom ◡𝐵)
5		df-rn 5700	. 2 ⊢ ran (𝐴 ∩ 𝐵) = dom ◡(𝐴 ∩ 𝐵)
6		df-rn 5700	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
7		df-rn 5700	. . 3 ⊢ ran 𝐵 = dom ◡𝐵
8	6, 7	ineq12i 4226	. 2 ⊢ (ran 𝐴 ∩ ran 𝐵) = (dom ◡𝐴 ∩ dom ◡𝐵)
9	4, 5, 8	3sstr4i 4039	1 ⊢ ran (𝐴 ∩ 𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵)

Syntax hints:   ∩ cin 3962   ⊆ wss 3963  ^◡ccnv 5688  dom cdm 5689  ran crn 5690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700

This theorem is referenced by:  inimass  6177  restutop  24262