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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 6323 | . . . 4 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹) | |
2 | 1 | 3ad2ant1 1126 | . . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → Fun 𝐹) |
3 | fnex 6846 | . . . 4 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ V) | |
4 | 3 | 3adant3 1125 | . . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝐹 ∈ V) |
5 | simp3 1131 | . . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑍 ∈ 𝑊) | |
6 | suppval1 7687 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹‘𝑖) ≠ 𝑍}) | |
7 | 2, 4, 5, 6 | syl3anc 1364 | . 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹‘𝑖) ≠ 𝑍}) |
8 | fndm 6325 | . . . 4 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋) | |
9 | 8 | 3ad2ant1 1126 | . . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → dom 𝐹 = 𝑋) |
10 | 9 | rabeqdv 3429 | . 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑖 ∈ dom 𝐹 ∣ (𝐹‘𝑖) ≠ 𝑍} = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍}) |
11 | 7, 10 | eqtrd 2831 | 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 1080 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 {crab 3109 Vcvv 3437 dom cdm 5443 Fun wfun 6219 Fn wfn 6220 ‘cfv 6225 (class class class)co 7016 supp csupp 7681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-ov 7019 df-oprab 7020 df-mpo 7021 df-supp 7682 |
This theorem is referenced by: elsuppfn 7689 cantnflem1 8998 fsuppmapnn0fiub0 13211 fsuppmapnn0ub 13213 mptnn0fsupp 13215 mptnn0fsuppr 13217 cicer 16905 mptscmfsupp0 19389 rrgsupp 19753 mhpinvcl 20022 frlmbas 20581 frlmssuvc2 20621 pmatcollpw2lem 21069 rrxmvallem 23690 fpwrelmapffslem 30156 fedgmullem2 30630 fsumcvg4 30810 fsumsupp0 41401 |
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