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Theorem cnvimainrn 7086
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 7083 . 2 (Fun 𝐹 → (𝐹 “ (ran 𝐹𝐴)) = ((𝐹 “ ran 𝐹) ∩ (𝐹𝐴)))
2 cnvimass 6101 . . . . 5 (𝐹𝐴) ⊆ dom 𝐹
3 cnvimarndm 6102 . . . . 5 (𝐹 “ ran 𝐹) = dom 𝐹
42, 3sseqtrri 4032 . . . 4 (𝐹𝐴) ⊆ (𝐹 “ ran 𝐹)
5 dfss2 3980 . . . 4 ((𝐹𝐴) ⊆ (𝐹 “ ran 𝐹) ↔ ((𝐹𝐴) ∩ (𝐹 “ ran 𝐹)) = (𝐹𝐴))
64, 5mpbi 230 . . 3 ((𝐹𝐴) ∩ (𝐹 “ ran 𝐹)) = (𝐹𝐴)
76ineqcomi 4218 . 2 ((𝐹 “ ran 𝐹) ∩ (𝐹𝐴)) = (𝐹𝐴)
81, 7eqtrdi 2790 1 (Fun 𝐹 → (𝐹 “ (ran 𝐹𝐴)) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  cin 3961  wss 3962  ccnv 5687  dom cdm 5688  ran crn 5689  cima 5691  Fun wfun 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-fun 6564
This theorem is referenced by:  fcoreslem1  47012
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