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| Mirrors > Home > MPE Home > Th. List > suppvalfng | Structured version Visualization version GIF version | ||
| Description: The value of the operation constructing the support of a function with a given domain. This version of suppvalfn 8149 assumes 𝐹 is a set rather than its domain 𝑋, avoiding ax-rep 5236. (Contributed by SN, 5-Aug-2024.) |
| Ref | Expression |
|---|---|
| suppvalfng | ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fnfun 6620 | . . 3 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹) | |
| 2 | suppval1 8147 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹‘𝑖) ≠ 𝑍}) | |
| 3 | 1, 2 | syl3an1 1163 | . 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹‘𝑖) ≠ 𝑍}) |
| 4 | fndm 6623 | . . . 4 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋) | |
| 5 | 4 | 3ad2ant1 1133 | . . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → dom 𝐹 = 𝑋) |
| 6 | 5 | rabeqdv 3424 | . 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑖 ∈ dom 𝐹 ∣ (𝐹‘𝑖) ≠ 𝑍} = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍}) |
| 7 | 3, 6 | eqtrd 2765 | 1 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 {crab 3408 dom cdm 5640 Fun wfun 6507 Fn wfn 6508 ‘cfv 6513 (class class class)co 7389 supp csupp 8141 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5253 ax-nul 5263 ax-pr 5389 ax-un 7713 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3756 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-br 5110 df-opab 5172 df-id 5535 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-iota 6466 df-fun 6515 df-fn 6516 df-fv 6521 df-ov 7392 df-oprab 7393 df-mpo 7394 df-supp 8142 |
| This theorem is referenced by: elsuppfng 8150 |
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