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Theorem elpreimad 7007
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 7006 . . 3 (𝐹 Fn 𝐴 → (𝐵 ∈ (𝐹𝐶) ↔ (𝐵𝐴 ∧ (𝐹𝐵) ∈ 𝐶)))
53, 4syl 17 . 2 (𝜑 → (𝐵 ∈ (𝐹𝐶) ↔ (𝐵𝐴 ∧ (𝐹𝐵) ∈ 𝐶)))
61, 2, 5mpbir2and 711 1 (𝜑𝐵 ∈ (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  ccnv 5631  cima 5635   Fn wfn 6489  cfv 6494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-fv 6502
This theorem is referenced by:  fpwwe2lem8  10571  evlslem3  21486  rhmpreimaidl  32091  rhmpreimaprmidl  32115  dimkerim  32213  smfsuplem1  45022
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