Theorem rninOLD 6143

Description: Obsolete version of rnin 6142 as of 10-Jun-2026. (Contributed by NM, 15-Sep-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem rninOLD

Step	Hyp	Ref	Expression
1		cnvin 6140	. . . 4 ⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵)
2	1	dmeqi 5893	. . 3 ⊢ dom ◡(𝐴 ∩ 𝐵) = dom (◡𝐴 ∩ ◡𝐵)
3		dmin 5900	. . 3 ⊢ dom (◡𝐴 ∩ ◡𝐵) ⊆ (dom ◡𝐴 ∩ dom ◡𝐵)
4	2, 3	eqsstri 3982	. 2 ⊢ dom ◡(𝐴 ∩ 𝐵) ⊆ (dom ◡𝐴 ∩ dom ◡𝐵)
5		df-rn 5671	. 2 ⊢ ran (𝐴 ∩ 𝐵) = dom ◡(𝐴 ∩ 𝐵)
6		df-rn 5671	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
7		df-rn 5671	. . 3 ⊢ ran 𝐵 = dom ◡𝐵
8	6, 7	ineq12i 4170	. 2 ⊢ (ran 𝐴 ∩ ran 𝐵) = (dom ◡𝐴 ∩ dom ◡𝐵)
9	4, 5, 8	3sstr4i 3987	1 ⊢ ran (𝐴 ∩ 𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵)

This proof depends on syntax axioms:   ∩ cin 3903   ⊆ wss 3904  ^◡ccnv 5659  dom cdm 5660  ran crn 5661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667  df-cnv 5668  df-dm 5670  df-rn 5671

This theorem is used by: (None)