Theorem inimass 5082

Description: The image of an intersection. (Contributed by Thierry Arnoux, 16-Dec-2017.)

Assertion

Ref	Expression
inimass	⊢ ((𝐴 ∩ 𝐵) “ 𝐶) ⊆ ((𝐴 “ 𝐶) ∩ (𝐵 “ 𝐶))

Proof of Theorem inimass

Step	Hyp	Ref	Expression
1		rnin 5075	. 2 ⊢ ran ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶)) ⊆ (ran (𝐴 ↾ 𝐶) ∩ ran (𝐵 ↾ 𝐶))
2		df-ima 4672	. . 3 ⊢ ((𝐴 ∩ 𝐵) “ 𝐶) = ran ((𝐴 ∩ 𝐵) ↾ 𝐶)
3		resindir 4958	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶))
4	3	rneqi 4890	. . 3 ⊢ ran ((𝐴 ∩ 𝐵) ↾ 𝐶) = ran ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶))
5	2, 4	eqtri 2214	. 2 ⊢ ((𝐴 ∩ 𝐵) “ 𝐶) = ran ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶))
6		df-ima 4672	. . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶)
7		df-ima 4672	. . 3 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶)
8	6, 7	ineq12i 3358	. 2 ⊢ ((𝐴 “ 𝐶) ∩ (𝐵 “ 𝐶)) = (ran (𝐴 ↾ 𝐶) ∩ ran (𝐵 ↾ 𝐶))
9	1, 5, 8	3sstr4i 3220	1 ⊢ ((𝐴 ∩ 𝐵) “ 𝐶) ⊆ ((𝐴 “ 𝐶) ∩ (𝐵 “ 𝐶))

Syntax hints:   ∩ cin 3152   ⊆ wss 3153  ran crn 4660   ↾ cres 4661   “ cima 4662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672

This theorem is referenced by: (None)