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Theorem fniniseg2 6296
 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 6295 . 2 (𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ {𝐵}})
2 fvex 6158 . . . . 5 (𝐹𝑥) ∈ V
32elsn 4163 . . . 4 ((𝐹𝑥) ∈ {𝐵} ↔ (𝐹𝑥) = 𝐵)
43a1i 11 . . 3 (𝑥𝐴 → ((𝐹𝑥) ∈ {𝐵} ↔ (𝐹𝑥) = 𝐵))
54rabbiia 3173 . 2 {𝑥𝐴 ∣ (𝐹𝑥) ∈ {𝐵}} = {𝑥𝐴 ∣ (𝐹𝑥) = 𝐵}
61, 5syl6eq 2671 1 (𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) = {𝑥𝐴 ∣ (𝐹𝑥) = 𝐵})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1480   ∈ wcel 1987  {crab 2911  {csn 4148  ◡ccnv 5073   “ cima 5077   Fn wfn 5842  ‘cfv 5847 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855 This theorem is referenced by:  idomrootle  37254  proot1hash  37259
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