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Mirrors > Home > MPE Home > Th. List > suppvalfn | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
suppvalfn | ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnfun 6650 | . . . 4 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹) | |
2 | 1 | 3ad2ant1 1130 | . . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → Fun 𝐹) |
3 | fnex 7224 | . . . 4 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ V) | |
4 | 3 | 3adant3 1129 | . . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝐹 ∈ V) |
5 | simp3 1135 | . . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑍 ∈ 𝑊) | |
6 | suppval1 8170 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹‘𝑖) ≠ 𝑍}) | |
7 | 2, 4, 5, 6 | syl3anc 1368 | . 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹‘𝑖) ≠ 𝑍}) |
8 | fndm 6653 | . . . 4 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋) | |
9 | 8 | 3ad2ant1 1130 | . . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → dom 𝐹 = 𝑋) |
10 | 9 | rabeqdv 3436 | . 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑖 ∈ dom 𝐹 ∣ (𝐹‘𝑖) ≠ 𝑍} = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍}) |
11 | 7, 10 | eqtrd 2766 | 1 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 {crab 3420 Vcvv 3463 dom cdm 5673 Fun wfun 6538 Fn wfn 6539 ‘cfv 6544 (class class class)co 7414 supp csupp 8164 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pr 5424 ax-un 7736 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7417 df-oprab 7418 df-mpo 7419 df-supp 8165 |
This theorem is referenced by: elsuppfn 8174 cantnflem1 9723 fsuppmapnn0fiub0 14005 fsuppmapnn0ub 14007 mptnn0fsupp 14009 mptnn0fsuppr 14011 cicer 17815 rrgsupp 20673 mptscmfsupp0 20897 frlmbas 21747 frlmssuvc2 21787 pmatcollpw2lem 22765 rrxmvallem 25418 fpwrelmapffslem 32644 fedgmullem2 33529 fsumcvg4 33776 fsuppind 42278 fsumsupp0 45233 |
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