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Theorem fncnvima2 6557
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 6555 . . 3 (𝐹 Fn 𝐴 → (𝑥 ∈ (𝐹𝐵) ↔ (𝑥𝐴 ∧ (𝐹𝑥) ∈ 𝐵)))
21abbi2dv 2926 . 2 (𝐹 Fn 𝐴 → (𝐹𝐵) = {𝑥 ∣ (𝑥𝐴 ∧ (𝐹𝑥) ∈ 𝐵)})
3 df-rab 3105 . 2 {𝑥𝐴 ∣ (𝐹𝑥) ∈ 𝐵} = {𝑥 ∣ (𝑥𝐴 ∧ (𝐹𝑥) ∈ 𝐵)}
42, 3syl6eqr 2858 1 (𝐹 Fn 𝐴 → (𝐹𝐵) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2156  {cab 2792  {crab 3100  ccnv 5310  cima 5314   Fn wfn 6092  cfv 6097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-fv 6105
This theorem is referenced by:  fniniseg2  6558  fncnvimaeqv  16960  r0cld  21751  iunpreima  29704  xppreima  29772  xpinpreima  30273  xpinpreima2  30274  orvcval2  30841  preimaiocmnf  40262  preimaicomnf  41398  smfresal  41471
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