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Theorem fncnvima2 5434
Description: Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
fncnvima2 (𝐹 Fn 𝐴 → (𝐹𝐵) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ 𝐵})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐵

Proof of Theorem fncnvima2
StepHypRef Expression
1 elpreima 5432 . . 3 (𝐹 Fn 𝐴 → (𝑥 ∈ (𝐹𝐵) ↔ (𝑥𝐴 ∧ (𝐹𝑥) ∈ 𝐵)))
21abbi2dv 2207 . 2 (𝐹 Fn 𝐴 → (𝐹𝐵) = {𝑥 ∣ (𝑥𝐴 ∧ (𝐹𝑥) ∈ 𝐵)})
3 df-rab 2369 . 2 {𝑥𝐴 ∣ (𝐹𝑥) ∈ 𝐵} = {𝑥 ∣ (𝑥𝐴 ∧ (𝐹𝑥) ∈ 𝐵)}
42, 3syl6eqr 2139 1 (𝐹 Fn 𝐴 → (𝐹𝐵) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ 𝐵})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1290  wcel 1439  {cab 2075  {crab 2364  ccnv 4450  cima 4454   Fn wfn 5023  cfv 5028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-fv 5036
This theorem is referenced by:  fniniseg2  5435  fnniniseg2  5436
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