Theorem rnin 6153

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 6151	. . . 4 ⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵)
2	1	dmeqi 5907	. . 3 ⊢ dom ◡(𝐴 ∩ 𝐵) = dom (◡𝐴 ∩ ◡𝐵)
3		dmin 5914	. . 3 ⊢ dom (◡𝐴 ∩ ◡𝐵) ⊆ (dom ◡𝐴 ∩ dom ◡𝐵)
4	2, 3	eqsstri 4011	. 2 ⊢ dom ◡(𝐴 ∩ 𝐵) ⊆ (dom ◡𝐴 ∩ dom ◡𝐵)
5		df-rn 5689	. 2 ⊢ ran (𝐴 ∩ 𝐵) = dom ◡(𝐴 ∩ 𝐵)
6		df-rn 5689	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
7		df-rn 5689	. . 3 ⊢ ran 𝐵 = dom ◡𝐵
8	6, 7	ineq12i 4208	. 2 ⊢ (ran 𝐴 ∩ ran 𝐵) = (dom ◡𝐴 ∩ dom ◡𝐵)
9	4, 5, 8	3sstr4i 4020	1 ⊢ ran (𝐴 ∩ 𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵)

Syntax hints:   ∩ cin 3943   ⊆ wss 3944  ^◡ccnv 5677  dom cdm 5678  ran crn 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689

This theorem is referenced by:  inimass  6161  restutop  24186