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Theorem fcoreslem1 47013
Description: Lemma 1 for fcores 47017. (Contributed by AV, 17-Sep-2024.)
Hypotheses
Ref Expression
fcores.f (𝜑𝐹:𝐴𝐵)
fcores.e 𝐸 = (ran 𝐹𝐶)
fcores.p 𝑃 = (𝐹𝐶)
Assertion
Ref Expression
fcoreslem1 (𝜑𝑃 = (𝐹𝐸))

Proof of Theorem fcoreslem1
StepHypRef Expression
1 fcores.f . . . . 5 (𝜑𝐹:𝐴𝐵)
21ffund 6741 . . . 4 (𝜑 → Fun 𝐹)
3 cnvimainrn 7087 . . . 4 (Fun 𝐹 → (𝐹 “ (ran 𝐹𝐶)) = (𝐹𝐶))
42, 3syl 17 . . 3 (𝜑 → (𝐹 “ (ran 𝐹𝐶)) = (𝐹𝐶))
54eqcomd 2741 . 2 (𝜑 → (𝐹𝐶) = (𝐹 “ (ran 𝐹𝐶)))
6 fcores.p . 2 𝑃 = (𝐹𝐶)
7 fcores.e . . 3 𝐸 = (ran 𝐹𝐶)
87imaeq2i 6078 . 2 (𝐹𝐸) = (𝐹 “ (ran 𝐹𝐶))
95, 6, 83eqtr4g 2800 1 (𝜑𝑃 = (𝐹𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  cin 3962  ccnv 5688  ran crn 5690  cima 5692  Fun wfun 6557  wf 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567
This theorem is referenced by:  fcoreslem2  47014
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