Theorem fniniseg2 7043

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 7042	. 2 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {𝐵}) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ {𝐵}})
2		fvex 6880	. . . 4 ⊢ (𝐹‘𝑥) ∈ V
3	2	elsn 4597	. . 3 ⊢ ((𝐹‘𝑥) ∈ {𝐵} ↔ (𝐹‘𝑥) = 𝐵)
4	3	rabbii 3419	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ {𝐵}} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝐵}
5	1, 4	eqtrdi 2813	1 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {𝐵}) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝐵})

Syntax hints:   → wi 4   = wceq 1560   ∈ wcel 2142  {crab 3414  {csn 4582  ^◡ccnv 5646   “ cima 5650   Fn wfn 6516  ‘cfv 6521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-fv 6529

This theorem is referenced by:  idomrootle  26233  qusker  33535  ply1dg1rt  33776  ply1mulrtss  33778  ply1annidllem  33998  algextdeglem6  34019  2sqr3minply  34077  cos9thpiminply  34085  aks6d1c6isolem3  42793  proot1hash  43772