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Theorem fcoreslem1 46508
Description: Lemma 1 for fcores 46512. (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 2731 . 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 2790 1 (𝜑𝑃 = (𝐹𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  cin 3938  ccnv 5671  ran crn 5673  cima 5675  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 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-fun 6545  df-fn 6546  df-f 6547
This theorem is referenced by:  fcoreslem2  46509
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