Theorem fniniseg2 7055

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 7054	. 2 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {𝐵}) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ {𝐵}})
2		fvex 6892	. . . 4 ⊢ (𝐹‘𝑥) ∈ V
3	2	elsn 4599	. . 3 ⊢ ((𝐹‘𝑥) ∈ {𝐵} ↔ (𝐹‘𝑥) = 𝐵)
4	3	rabbii 3417	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ {𝐵}} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝐵}
5	1, 4	eqtrdi 2811	1 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {𝐵}) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝐵})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {crab 3412  {csn 4584  ^◡ccnv 5654   “ cima 5658   Fn wfn 6528  ‘cfv 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-fv 6541

This theorem is used by:  idomrootle  26399  qusker  33790  ply1dg1rt  33991  ply1mulrtss  33993  ply1annidllem  34212  algextdeglem6  34233  2sqr3minply  34291  cos9thpiminply  34299  aks6d1c6isolem3  43043  proot1hash  44037