Theorem fncnvima2 7056

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 7053	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ (◡𝐹 “ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ 𝐵)))
2	1	eqabdv 2896	. 2 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ 𝐵)})
3		df-rab 3417	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ 𝐵)}
4	2, 3	eqtr4di 2816	1 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ 𝐵) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ 𝐵})

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {cab 2741  {crab 3416  ^◡ccnv 5660   “ cima 5664   Fn wfn 6531  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544

This theorem is referenced by:  fniniseg2  7057  fncnvimaeqv  18171  rngqiprngimf1  21440  r0cld  23895  iunpreima  32909  xppreima  32990  xpinpreima  34296  xpinpreima2  34297  orvcval2  34849  preimaiocmnf  46276  preimaicomnf  47425  smfresal  47502