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Theorem fcoreslem3 47015
Description: Lemma 3 for fcores 47017. (Contributed by AV, 13-Sep-2024.)
Hypotheses
Ref Expression
fcores.f (𝜑𝐹:𝐴𝐵)
fcores.e 𝐸 = (ran 𝐹𝐶)
fcores.p 𝑃 = (𝐹𝐶)
fcores.x 𝑋 = (𝐹𝑃)
Assertion
Ref Expression
fcoreslem3 (𝜑𝑋:𝑃onto𝐸)

Proof of Theorem fcoreslem3
StepHypRef Expression
1 fcores.f . . . 4 (𝜑𝐹:𝐴𝐵)
21ffnd 6738 . . 3 (𝜑𝐹 Fn 𝐴)
3 fcores.e . . . 4 𝐸 = (ran 𝐹𝐶)
43a1i 11 . . 3 (𝜑𝐸 = (ran 𝐹𝐶))
5 fcores.p . . . 4 𝑃 = (𝐹𝐶)
65a1i 11 . . 3 (𝜑𝑃 = (𝐹𝐶))
72, 4, 6rescnvimafod 7093 . 2 (𝜑 → (𝐹𝑃):𝑃onto𝐸)
8 fcores.x . . 3 𝑋 = (𝐹𝑃)
9 foeq1 6817 . . 3 (𝑋 = (𝐹𝑃) → (𝑋:𝑃onto𝐸 ↔ (𝐹𝑃):𝑃onto𝐸))
108, 9mp1i 13 . 2 (𝜑 → (𝑋:𝑃onto𝐸 ↔ (𝐹𝑃):𝑃onto𝐸))
117, 10mpbird 257 1 (𝜑𝑋:𝑃onto𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  cin 3962  ccnv 5688  ran crn 5690  cres 5691  cima 5692  wf 6559  ontowfo 6561
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-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-sb 2063  df-mo 2538  df-eu 2567  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  df-fo 6569
This theorem is referenced by:  fcoreslem4  47016  fcores  47017  fcoresf1lem  47018  fcoresf1  47019  fcoresfo  47021  fcoresfob  47022
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