Theorem fniinfv 6399

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		fnrnfv 6384	. . 3 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
2	1	inteqd 4616	. 2 ⊢ (𝐹 Fn 𝐴 → ∩ ran 𝐹 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
3		fvex 6342	. . 3 ⊢ (𝐹‘𝑥) ∈ V
4	3	dfiin2 4689	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}
5	2, 4	syl6reqr 2824	1 ⊢ (𝐹 Fn 𝐴 → ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ ran 𝐹)

Syntax hints:   → wi 4   = wceq 1631  {cab 2757  ∃wrex 3062  ∩ cint 4611  ∩ ciin 4655  ran crn 5250   Fn wfn 6026  ‘cfv 6031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039

This theorem is referenced by:  firest  16301  pnrmopn  21368  txtube  21664  bcth3  23347  diaintclN  36868  dibintclN  36977  dihintcl  37154  imaiinfv  37782