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Theorem fncnvima2 6481
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 6479 . . 3 (𝐹 Fn 𝐴 → (𝑥 ∈ (𝐹𝐵) ↔ (𝑥𝐴 ∧ (𝐹𝑥) ∈ 𝐵)))
21abbi2dv 2891 . 2 (𝐹 Fn 𝐴 → (𝐹𝐵) = {𝑥 ∣ (𝑥𝐴 ∧ (𝐹𝑥) ∈ 𝐵)})
3 df-rab 3070 . 2 {𝑥𝐴 ∣ (𝐹𝑥) ∈ 𝐵} = {𝑥 ∣ (𝑥𝐴 ∧ (𝐹𝑥) ∈ 𝐵)}
42, 3syl6eqr 2823 1 (𝐹 Fn 𝐴 → (𝐹𝐵) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {cab 2757  {crab 3065  ccnv 5248  cima 5252   Fn wfn 6024  cfv 6029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-fv 6037
This theorem is referenced by:  fniniseg2  6482  fncnvimaeqv  16963  r0cld  21758  iunpreima  29717  xppreima  29785  xpinpreima  30288  xpinpreima2  30289  orvcval2  30856  preimaiocmnf  40303  preimaicomnf  41439  smfresal  41512
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