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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 6446 | . . . 4 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹) | |
2 | 1 | 3ad2ant1 1125 | . . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → Fun 𝐹) |
3 | fnex 6971 | . . . 4 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ V) | |
4 | 3 | 3adant3 1124 | . . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝐹 ∈ V) |
5 | simp3 1130 | . . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑍 ∈ 𝑊) | |
6 | suppval1 7825 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹‘𝑖) ≠ 𝑍}) | |
7 | 2, 4, 5, 6 | syl3anc 1363 | . 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹‘𝑖) ≠ 𝑍}) |
8 | fndm 6448 | . . . 4 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋) | |
9 | 8 | 3ad2ant1 1125 | . . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → dom 𝐹 = 𝑋) |
10 | 9 | rabeqdv 3482 | . 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑖 ∈ dom 𝐹 ∣ (𝐹‘𝑖) ≠ 𝑍} = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍}) |
11 | 7, 10 | eqtrd 2853 | 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 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 {crab 3139 Vcvv 3492 dom cdm 5548 Fun wfun 6342 Fn wfn 6343 ‘cfv 6348 (class class class)co 7145 supp csupp 7819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-supp 7820 |
This theorem is referenced by: elsuppfn 7827 cantnflem1 9140 fsuppmapnn0fiub0 13349 fsuppmapnn0ub 13351 mptnn0fsupp 13353 mptnn0fsuppr 13355 cicer 17064 mptscmfsupp0 19628 rrgsupp 19992 mhpinvcl 20267 frlmbas 20827 frlmssuvc2 20867 pmatcollpw2lem 21313 rrxmvallem 23934 fpwrelmapffslem 30394 fedgmullem2 30925 fsumcvg4 31092 fsumsupp0 41735 |
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