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Theorem fniinfv 5152
 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 (𝐹x) = ran 𝐹)
Distinct variable groups:   x,A   x,𝐹

Proof of Theorem fniinfv
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 funfvex 5113 . . . . 5 ((Fun 𝐹 x dom 𝐹) → (𝐹x) V)
21funfni 4921 . . . 4 ((𝐹 Fn A x A) → (𝐹x) V)
32ralrimiva 2366 . . 3 (𝐹 Fn Ax A (𝐹x) V)
4 dfiin2g 3660 . . 3 (x A (𝐹x) V → x A (𝐹x) = {yx A y = (𝐹x)})
53, 4syl 14 . 2 (𝐹 Fn A x A (𝐹x) = {yx A y = (𝐹x)})
6 fnrnfv 5141 . . 3 (𝐹 Fn A → ran 𝐹 = {yx A y = (𝐹x)})
76inteqd 3590 . 2 (𝐹 Fn A ran 𝐹 = {yx A y = (𝐹x)})
85, 7eqtr4d 2053 1 (𝐹 Fn A x A (𝐹x) = ran 𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1226   ∈ wcel 1370  {cab 2004  ∀wral 2280  ∃wrex 2281  Vcvv 2531  ∩ cint 3585  ∩ ciin 3628  ran crn 4269   Fn wfn 4820  ‘cfv 4825 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-int 3586  df-iin 3630  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-iota 4790  df-fun 4827  df-fn 4828  df-fv 4833 This theorem is referenced by: (None)
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