Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fcoreslem1 Structured version   Visualization version   GIF version

Theorem fcoreslem1 47075
Description: Lemma 1 for fcores 47079. (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 6740 . . . 4 (𝜑 → Fun 𝐹)
3 cnvimainrn 7087 . . . 4 (Fun 𝐹 → (𝐹 “ (ran 𝐹𝐶)) = (𝐹𝐶))
42, 3syl 17 . . 3 (𝜑 → (𝐹 “ (ran 𝐹𝐶)) = (𝐹𝐶))
54eqcomd 2743 . 2 (𝜑 → (𝐹𝐶) = (𝐹 “ (ran 𝐹𝐶)))
6 fcores.p . 2 𝑃 = (𝐹𝐶)
7 fcores.e . . 3 𝐸 = (ran 𝐹𝐶)
87imaeq2i 6076 . 2 (𝐹𝐸) = (𝐹 “ (ran 𝐹𝐶))
95, 6, 83eqtr4g 2802 1 (𝜑𝑃 = (𝐹𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  cin 3950  ccnv 5684  ran crn 5686  cima 5688  Fun wfun 6555  wf 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565
This theorem is referenced by:  fcoreslem2  47076
  Copyright terms: Public domain W3C validator