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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 7447 | . . 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 5462 | . . . . 5 ⊢ (𝑥 = 𝑋 → dom 𝑥 = dom 𝑋) | |
4 | 3 | adantr 466 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → dom 𝑥 = dom 𝑋) |
5 | imaeq1 5602 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 “ {𝑖}) = (𝑋 “ {𝑖})) | |
6 | 5 | adantr 466 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → (𝑥 “ {𝑖}) = (𝑋 “ {𝑖})) |
7 | sneq 4326 | . . . . . 6 ⊢ (𝑧 = 𝑍 → {𝑧} = {𝑍}) | |
8 | 7 | adantl 467 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → {𝑧} = {𝑍}) |
9 | 6, 8 | neeq12d 3004 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → ((𝑥 “ {𝑖}) ≠ {𝑧} ↔ (𝑋 “ {𝑖}) ≠ {𝑍})) |
10 | 4, 9 | rabeqbidv 3345 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑧 = 𝑍) → {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}}) |
11 | 10 | adantl 467 | . 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝑥 = 𝑋 ∧ 𝑧 = 𝑍)) → {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}} = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}}) |
12 | elex 3364 | . . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V) | |
13 | 12 | adantr 466 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑋 ∈ V) |
14 | elex 3364 | . . 3 ⊢ (𝑍 ∈ 𝑊 → 𝑍 ∈ V) | |
15 | 14 | adantl 467 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝑍 ∈ V) |
16 | dmexg 7244 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → dom 𝑋 ∈ V) | |
17 | 16 | adantr 466 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → dom 𝑋 ∈ V) |
18 | rabexg 4945 | . . 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 | ovmpt2d 6935 | 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 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 {crab 3065 Vcvv 3351 {csn 4316 dom cdm 5249 “ cima 5252 (class class class)co 6793 ↦ cmpt2 6795 supp csupp 7446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fv 6039 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-supp 7447 |
This theorem is referenced by: suppvalbr 7450 supp0 7451 suppval1 7452 suppssdm 7459 suppsnop 7460 ressuppss 7465 ressuppssdif 7467 |
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