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Theorem inimass 4683
 Description: The image of an intersection (Contributed by Thierry Arnoux, 16-Dec-2017.)
Assertion
Ref Expression
inimass ((AB) “ 𝐶) ⊆ ((A𝐶) ∩ (B𝐶))

Proof of Theorem inimass
StepHypRef Expression
1 rnin 4676 . 2 ran ((A𝐶) ∩ (B𝐶)) ⊆ (ran (A𝐶) ∩ ran (B𝐶))
2 df-ima 4301 . . 3 ((AB) “ 𝐶) = ran ((AB) ↾ 𝐶)
3 resindir 4571 . . . 4 ((AB) ↾ 𝐶) = ((A𝐶) ∩ (B𝐶))
43rneqi 4505 . . 3 ran ((AB) ↾ 𝐶) = ran ((A𝐶) ∩ (B𝐶))
52, 4eqtri 2057 . 2 ((AB) “ 𝐶) = ran ((A𝐶) ∩ (B𝐶))
6 df-ima 4301 . . 3 (A𝐶) = ran (A𝐶)
7 df-ima 4301 . . 3 (B𝐶) = ran (B𝐶)
86, 7ineq12i 3130 . 2 ((A𝐶) ∩ (B𝐶)) = (ran (A𝐶) ∩ ran (B𝐶))
91, 5, 83sstr4i 2978 1 ((AB) “ 𝐶) ⊆ ((A𝐶) ∩ (B𝐶))
 Colors of variables: wff set class Syntax hints:   ∩ cin 2910   ⊆ wss 2911  ran crn 4289   ↾ cres 4290   “ cima 4291 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301 This theorem is referenced by: (None)
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