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Theorem elpreimad 7078
Description: Membership in the preimage of a set under a function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
elpreimad.f (𝜑𝐹 Fn 𝐴)
elpreimad.b (𝜑𝐵𝐴)
elpreimad.c (𝜑 → (𝐹𝐵) ∈ 𝐶)
Assertion
Ref Expression
elpreimad (𝜑𝐵 ∈ (𝐹𝐶))

Proof of Theorem elpreimad
StepHypRef Expression
1 elpreimad.b . 2 (𝜑𝐵𝐴)
2 elpreimad.c . 2 (𝜑 → (𝐹𝐵) ∈ 𝐶)
3 elpreimad.f . . 3 (𝜑𝐹 Fn 𝐴)
4 elpreima 7077 . . 3 (𝐹 Fn 𝐴 → (𝐵 ∈ (𝐹𝐶) ↔ (𝐵𝐴 ∧ (𝐹𝐵) ∈ 𝐶)))
53, 4syl 17 . 2 (𝜑 → (𝐵 ∈ (𝐹𝐶) ↔ (𝐵𝐴 ∧ (𝐹𝐵) ∈ 𝐶)))
61, 2, 5mpbir2and 713 1 (𝜑𝐵 ∈ (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2105  ccnv 5687  cima 5691   Fn wfn 6557  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570
This theorem is referenced by:  fpwwe2lem8  10675  rhmpreimaidl  21304  evlslem3  22121  rhmpreimaprmidl  33458  ply1degltel  33594  ply1degleel  33595  ply1degltlss  33596  ply1degltdimlem  33649  ply1degltdim  33650  dimkerim  33654  lvecendof1f1o  33660  zndvdchrrhm  41952  smfsuplem1  46766
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