ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fncnvima2 Structured version   GIF version

Theorem fncnvima2 5231
Description: Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
fncnvima2 (𝐹 Fn A → (𝐹B) = {x A ∣ (𝐹x) B})
Distinct variable groups:   x,A   x,𝐹   x,B

Proof of Theorem fncnvima2
StepHypRef Expression
1 elpreima 5229 . . 3 (𝐹 Fn A → (x (𝐹B) ↔ (x A (𝐹x) B)))
21abbi2dv 2153 . 2 (𝐹 Fn A → (𝐹B) = {x ∣ (x A (𝐹x) B)})
3 df-rab 2309 . 2 {x A ∣ (𝐹x) B} = {x ∣ (x A (𝐹x) B)}
42, 3syl6eqr 2087 1 (𝐹 Fn A → (𝐹B) = {x A ∣ (𝐹x) B})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  {cab 2023  {crab 2304  ccnv 4287  cima 4291   Fn wfn 4840  cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  fniniseg2  5232  fnniniseg2  5233
  Copyright terms: Public domain W3C validator