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Mirrors > Home > MPE Home > Th. List > suppval | Structured version Visualization version GIF version |
Description: The value of the operation constructing the support of a function. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 6-Apr-2019.) |
Ref | Expression |
---|---|
suppval | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-supp 7633 | . . 3 ⊢ supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}) | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})) |
3 | dmeq 5619 | . . . . 5 ⊢ (𝑥 = 𝑋 → dom 𝑥 = dom 𝑋) | |
4 | 3 | adantr 473 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → dom 𝑥 = dom 𝑋) |
5 | imaeq1 5763 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 “ {𝑖}) = (𝑋 “ {𝑖})) | |
6 | 5 | adantr 473 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → (𝑥 “ {𝑖}) = (𝑋 “ {𝑖})) |
7 | sneq 4446 | . . . . . 6 ⊢ (𝑧 = 𝑍 → {𝑧} = {𝑍}) | |
8 | 7 | adantl 474 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → {𝑧} = {𝑍}) |
9 | 6, 8 | neeq12d 3023 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → ((𝑥 “ {𝑖}) ≠ {𝑧} ↔ (𝑋 “ {𝑖}) ≠ {𝑍})) |
10 | 4, 9 | rabeqbidv 3403 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}}) |
11 | 10 | adantl 474 | . 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝑥 = 𝑋 ∧ 𝑧 = 𝑍)) → {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}}) |
12 | elex 3428 | . . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V) | |
13 | 12 | adantr 473 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑋 ∈ V) |
14 | elex 3428 | . . 3 ⊢ (𝑍 ∈ 𝑊 → 𝑍 ∈ V) | |
15 | 14 | adantl 474 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑍 ∈ V) |
16 | dmexg 7427 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → dom 𝑋 ∈ V) | |
17 | 16 | adantr 473 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → dom 𝑋 ∈ V) |
18 | rabexg 5087 | . . 3 ⊢ (dom 𝑋 ∈ V → {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}} ∈ V) | |
19 | 17, 18 | syl 17 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}} ∈ V) |
20 | 2, 11, 13, 15, 19 | ovmpod 7117 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ≠ wne 2962 {crab 3087 Vcvv 3410 {csn 4436 dom cdm 5404 “ cima 5407 (class class class)co 6975 ∈ cmpo 6977 supp csupp 7632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pr 5183 ax-un 7278 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-rab 3092 df-v 3412 df-sbc 3677 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-br 4927 df-opab 4989 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-iota 6150 df-fun 6188 df-fv 6194 df-ov 6978 df-oprab 6979 df-mpo 6980 df-supp 7633 |
This theorem is referenced by: suppvalbr 7636 supp0 7637 suppval1 7638 suppssdm 7645 suppsnop 7646 ressuppss 7651 ressuppssdif 7653 |
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