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Theorem fncnvima2 6827
 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 6824 . . 3 (𝐹 Fn 𝐴 → (𝑥 ∈ (𝐹𝐵) ↔ (𝑥𝐴 ∧ (𝐹𝑥) ∈ 𝐵)))
21abbi2dv 2955 . 2 (𝐹 Fn 𝐴 → (𝐹𝐵) = {𝑥 ∣ (𝑥𝐴 ∧ (𝐹𝑥) ∈ 𝐵)})
3 df-rab 3152 . 2 {𝑥𝐴 ∣ (𝐹𝑥) ∈ 𝐵} = {𝑥 ∣ (𝑥𝐴 ∧ (𝐹𝑥) ∈ 𝐵)}
42, 3syl6eqr 2879 1 (𝐹 Fn 𝐴 → (𝐹𝐵) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ 𝐵})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   = wceq 1530   ∈ wcel 2107  {cab 2804  {crab 3147  ◡ccnv 5553   “ cima 5557   Fn wfn 6347  ‘cfv 6352 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-fv 6360 This theorem is referenced by:  fniniseg2  6828  suppcofnd  7862  fncnvimaeqv  17360  r0cld  22265  iunpreima  30234  xppreima  30312  xpinpreima  31038  xpinpreima2  31039  orvcval2  31605  preimaiocmnf  41705  preimaicomnf  42859  smfresal  42932
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