Theorem fniinfv 6913

Description: The indexed intersection of a function's values is the intersection of its range. (Contributed by NM, 20-Oct-2005.)

Assertion

Ref	Expression
fniinfv	⊢ (𝐹 Fn 𝐴 → ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ ran 𝐹)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem fniinfv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6848	. . 3 ⊢ (𝐹‘𝑥) ∈ V
2	1	dfiin2 4989	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}
3		fnrnfv 6894	. . 3 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
4	3	inteqd 4908	. 2 ⊢ (𝐹 Fn 𝐴 → ∩ ran 𝐹 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
5	2, 4	eqtr4id 2791	1 ⊢ (𝐹 Fn 𝐴 → ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ ran 𝐹)

Syntax hints:   → wi 4   = wceq 1542  {cab 2715  ∃wrex 3061  ∩ cint 4903  ∩ ciin 4948  ran crn 5626   Fn wfn 6488  ‘cfv 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iin 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fn 6496  df-fv 6501

This theorem is referenced by:  firest  17357  pnrmopn  23292  txtube  23589  bcth3  25292  diaintclN  41397  dibintclN  41506  dihintcl  41683  imaiinfv  43013