Theorem fncnvima2 7033

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 7030	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ (◡𝐹 “ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ 𝐵)))
2	1	eqabdv 2861	. 2 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ 𝐵)})
3		df-rab 3406	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ 𝐵)}
4	2, 3	eqtr4di 2782	1 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ 𝐵) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ 𝐵})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  {crab 3405  ^◡ccnv 5637   “ cima 5641   Fn wfn 6506  ‘cfv 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-fv 6519

This theorem is referenced by:  fniniseg2  7034  fncnvimaeqv  18081  rngqiprngimf1  21210  r0cld  23625  iunpreima  32493  xppreima  32569  xpinpreima  33896  xpinpreima2  33897  orvcval2  34450  preimaiocmnf  45558  preimaicomnf  46709  smfresal  46786