Theorem fncnvima2 7003

Description: Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Assertion

Ref	Expression
fncnvima2	⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ 𝐵) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ 𝐵})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐵

Proof of Theorem fncnvima2

Step	Hyp	Ref	Expression
1		elpreima 7000	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ (◡𝐹 “ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ 𝐵)))
2	1	eqabdv 2866	. 2 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ 𝐵)})
3		df-rab 3397	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ 𝐵)}
4	2, 3	eqtr4di 2786	1 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ 𝐵) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ 𝐵})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2711  {crab 3396  ^◡ccnv 5620   “ cima 5624   Fn wfn 6484  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-fv 6497

This theorem is referenced by:  fniniseg2  7004  fncnvimaeqv  18034  rngqiprngimf1  21246  r0cld  23673  iunpreima  32565  xppreima  32649  xpinpreima  33991  xpinpreima2  33992  orvcval2  34544  preimaiocmnf  45722  preimaicomnf  46871  smfresal  46948