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Theorem fniniseg2 7063
Description: Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
fniniseg2 (𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) = {𝑥𝐴 ∣ (𝐹𝑥) = 𝐵})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐵

Proof of Theorem fniniseg2
StepHypRef Expression
1 fncnvima2 7062 . 2 (𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ {𝐵}})
2 fvex 6904 . . . 4 (𝐹𝑥) ∈ V
32elsn 4643 . . 3 ((𝐹𝑥) ∈ {𝐵} ↔ (𝐹𝑥) = 𝐵)
43rabbii 3437 . 2 {𝑥𝐴 ∣ (𝐹𝑥) ∈ {𝐵}} = {𝑥𝐴 ∣ (𝐹𝑥) = 𝐵}
51, 4eqtrdi 2787 1 (𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) = {𝑥𝐴 ∣ (𝐹𝑥) = 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  {crab 3431  {csn 4628  ccnv 5675  cima 5679   Fn wfn 6538  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551
This theorem is referenced by:  qusker  32900  ply1annidllem  33217  algextdeglem6  33233  idomrootle  42400  proot1hash  42405
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