Theorem suppvalfn 7189

Description: The value of the operation constructing the support of a function with a given domain. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 22-Apr-2019.)

Assertion

Ref	Expression
suppvalfn	⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍})

Distinct variable groups:   𝑖,𝑉   𝑖,𝑊   𝑖,𝑋   𝑖,𝑍   𝑖,𝐹

Proof of Theorem suppvalfn

Step	Hyp	Ref	Expression
1		fnfun 5902	. . . 4 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
2	1	3ad2ant1 1075	. . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → Fun 𝐹)
3		fnex 6386	. . . 4 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ V)
4	3	3adant3 1074	. . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝐹 ∈ V)
5		simp3 1056	. . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑍 ∈ 𝑊)
6		suppval1 7188	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹‘𝑖) ≠ 𝑍})
7	2, 4, 5, 6	syl3anc 1318	. 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹‘𝑖) ≠ 𝑍})
8		fndm 5904	. . . 4 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
9	8	3ad2ant1 1075	. . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → dom 𝐹 = 𝑋)
10		rabeq 3166	. . 3 ⊢ (dom 𝐹 = 𝑋 → {𝑖 ∈ dom 𝐹 ∣ (𝐹‘𝑖) ≠ 𝑍} = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍})
11	9, 10	syl 17	. 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑖 ∈ dom 𝐹 ∣ (𝐹‘𝑖) ≠ 𝑍} = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍})
12	7, 11	eqtrd 2644	1 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍})

Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900  Vcvv 3173  dom cdm 5038  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549   supp csupp 7182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-supp 7183

This theorem is referenced by:  elsuppfn  7190  cantnflem1  8469  fsuppmapnn0fiub0  12655  fsuppmapnn0ub  12657  mptnn0fsupp  12659  mptnn0fsuppr  12661  cicer  16289  mptscmfsupp0  18751  rrgsupp  19112  frlmbas  19918  frlmssuvc2  19953  pmatcollpw2lem  20401  rrxmvallem  22995  fpwrelmapffslem  28895  fsumcvg4  29324  fsumsupp0  38645