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Theorem cnvimainrn 7087
Description: The preimage of the intersection of the range of a class and a class 𝐴 is the preimage of the class 𝐴. (Contributed by AV, 17-Sep-2024.)
Assertion
Ref Expression
cnvimainrn (Fun 𝐹 → (𝐹 “ (ran 𝐹𝐴)) = (𝐹𝐴))

Proof of Theorem cnvimainrn
StepHypRef Expression
1 inpreima 7084 . 2 (Fun 𝐹 → (𝐹 “ (ran 𝐹𝐴)) = ((𝐹 “ ran 𝐹) ∩ (𝐹𝐴)))
2 cnvimass 6100 . . . . 5 (𝐹𝐴) ⊆ dom 𝐹
3 cnvimarndm 6101 . . . . 5 (𝐹 “ ran 𝐹) = dom 𝐹
42, 3sseqtrri 4033 . . . 4 (𝐹𝐴) ⊆ (𝐹 “ ran 𝐹)
5 dfss2 3969 . . . 4 ((𝐹𝐴) ⊆ (𝐹 “ ran 𝐹) ↔ ((𝐹𝐴) ∩ (𝐹 “ ran 𝐹)) = (𝐹𝐴))
64, 5mpbi 230 . . 3 ((𝐹𝐴) ∩ (𝐹 “ ran 𝐹)) = (𝐹𝐴)
76ineqcomi 4211 . 2 ((𝐹 “ ran 𝐹) ∩ (𝐹𝐴)) = (𝐹𝐴)
81, 7eqtrdi 2793 1 (Fun 𝐹 → (𝐹 “ (ran 𝐹𝐴)) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  cin 3950  wss 3951  ccnv 5684  dom cdm 5685  ran crn 5686  cima 5688  Fun wfun 6555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563
This theorem is referenced by:  fcoreslem1  47075
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