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Theorem fniniseg2 6819
 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 6818 . 2 (𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ {𝐵}})
2 fvex 6668 . . . 4 (𝐹𝑥) ∈ V
32elsn 4543 . . 3 ((𝐹𝑥) ∈ {𝐵} ↔ (𝐹𝑥) = 𝐵)
43rabbii 3421 . 2 {𝑥𝐴 ∣ (𝐹𝑥) ∈ {𝐵}} = {𝑥𝐴 ∣ (𝐹𝑥) = 𝐵}
51, 4eqtrdi 2849 1 (𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) = {𝑥𝐴 ∣ (𝐹𝑥) = 𝐵})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {crab 3110  {csn 4528  ◡ccnv 5522   “ cima 5526   Fn wfn 6327  ‘cfv 6332 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-fv 6340 This theorem is referenced by:  qusker  31018  idomrootle  40310  proot1hash  40315
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