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Theorem fncnvima2 6232
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 6230 . . 3 (𝐹 Fn 𝐴 → (𝑥 ∈ (𝐹𝐵) ↔ (𝑥𝐴 ∧ (𝐹𝑥) ∈ 𝐵)))
21abbi2dv 2728 . 2 (𝐹 Fn 𝐴 → (𝐹𝐵) = {𝑥 ∣ (𝑥𝐴 ∧ (𝐹𝑥) ∈ 𝐵)})
3 df-rab 2904 . 2 {𝑥𝐴 ∣ (𝐹𝑥) ∈ 𝐵} = {𝑥 ∣ (𝑥𝐴 ∧ (𝐹𝑥) ∈ 𝐵)}
42, 3syl6eqr 2661 1 (𝐹 Fn 𝐴 → (𝐹𝐵) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  {cab 2595  {crab 2899  ccnv 5027  cima 5031   Fn wfn 5785  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-fv 5798
This theorem is referenced by:  fniniseg2  6233  fncnvimaeqv  16532  r0cld  21299  iunpreima  28599  xppreima  28663  xpinpreima  29114  xpinpreima2  29115  orvcval2  29681  preimaicomnf  39423  smfresal  39497
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