Theorem fniniseg2 7095

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}) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝐵})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐵

Proof of Theorem fniniseg2

Step	Hyp	Ref	Expression
1		fncnvima2 7094	. 2 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {𝐵}) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ {𝐵}})
2		fvex 6933	. . . 4 ⊢ (𝐹‘𝑥) ∈ V
3	2	elsn 4663	. . 3 ⊢ ((𝐹‘𝑥) ∈ {𝐵} ↔ (𝐹‘𝑥) = 𝐵)
4	3	rabbii 3449	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ {𝐵}} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝐵}
5	1, 4	eqtrdi 2796	1 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {𝐵}) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝐵})

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  {crab 3443  {csn 4648  ^◡ccnv 5699   “ cima 5703   Fn wfn 6568  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581

This theorem is referenced by:  idomrootle  26232  qusker  33342  ply1dg1rt  33569  ply1mulrtss  33571  ply1annidllem  33694  algextdeglem6  33713  2sqr3minply  33738  aks6d1c6isolem3  42133  proot1hash  43156