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Theorem fcoreslem1 46436
Description: Lemma 1 for fcores 46440. (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 6721 . . . 4 (𝜑 → Fun 𝐹)
3 cnvimainrn 7071 . . . 4 (Fun 𝐹 → (𝐹 “ (ran 𝐹𝐶)) = (𝐹𝐶))
42, 3syl 17 . . 3 (𝜑 → (𝐹 “ (ran 𝐹𝐶)) = (𝐹𝐶))
54eqcomd 2734 . 2 (𝜑 → (𝐹𝐶) = (𝐹 “ (ran 𝐹𝐶)))
6 fcores.p . 2 𝑃 = (𝐹𝐶)
7 fcores.e . . 3 𝐸 = (ran 𝐹𝐶)
87imaeq2i 6056 . 2 (𝐹𝐸) = (𝐹 “ (ran 𝐹𝐶))
95, 6, 83eqtr4g 2793 1 (𝜑𝑃 = (𝐹𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  cin 3944  ccnv 5672  ran crn 5674  cima 5676  Fun wfun 6537  wf 6539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5144  df-opab 5206  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-fun 6545  df-fn 6546  df-f 6547
This theorem is referenced by:  fcoreslem2  46437
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