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Theorem fniniseg2 5210
 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 ∣ (𝐹x) = B})
Distinct variable groups:   x,A   x,𝐹   x,B

Proof of Theorem fniniseg2
StepHypRef Expression
1 fncnvima2 5209 . 2 (𝐹 Fn A → (𝐹 “ {B}) = {x A ∣ (𝐹x) {B}})
2 funfvex 5113 . . . . 5 ((Fun 𝐹 x dom 𝐹) → (𝐹x) V)
3 elsncg 3368 . . . . 5 ((𝐹x) V → ((𝐹x) {B} ↔ (𝐹x) = B))
42, 3syl 14 . . . 4 ((Fun 𝐹 x dom 𝐹) → ((𝐹x) {B} ↔ (𝐹x) = B))
54funfni 4921 . . 3 ((𝐹 Fn A x A) → ((𝐹x) {B} ↔ (𝐹x) = B))
65rabbidva 2522 . 2 (𝐹 Fn A → {x A ∣ (𝐹x) {B}} = {x A ∣ (𝐹x) = B})
71, 6eqtrd 2050 1 (𝐹 Fn A → (𝐹 “ {B}) = {x A ∣ (𝐹x) = B})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1226   ∈ wcel 1370  {crab 2284  Vcvv 2531  {csn 3346  ◡ccnv 4267  dom cdm 4268   “ cima 4271  Fun wfun 4819   Fn wfn 4820  ‘cfv 4825 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-rab 2289  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-fv 4833 This theorem is referenced by: (None)
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