![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > suppss2f | Structured version Visualization version GIF version |
Description: Show that the support of a function is contained in a set. (Contributed by Thierry Arnoux, 22-Jun-2017.) (Revised by AV, 1-Sep-2020.) |
Ref | Expression |
---|---|
suppss2f.p | ⊢ Ⅎ𝑘𝜑 |
suppss2f.a | ⊢ Ⅎ𝑘𝐴 |
suppss2f.w | ⊢ Ⅎ𝑘𝑊 |
suppss2f.n | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) |
suppss2f.v | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
suppss2f | ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suppss2f.a | . . . 4 ⊢ Ⅎ𝑘𝐴 | |
2 | nfcv 2899 | . . . 4 ⊢ Ⅎ𝑙𝐴 | |
3 | nfcv 2899 | . . . 4 ⊢ Ⅎ𝑙𝐵 | |
4 | nfcsb1v 3915 | . . . 4 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐵 | |
5 | csbeq1a 3904 | . . . 4 ⊢ (𝑘 = 𝑙 → 𝐵 = ⦋𝑙 / 𝑘⦌𝐵) | |
6 | 1, 2, 3, 4, 5 | cbvmptf 5251 | . . 3 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵) |
7 | 6 | oveq1i 7424 | . 2 ⊢ ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) = ((𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵) supp 𝑍) |
8 | suppss2f.n | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) | |
9 | 8 | sbt 2062 | . . . 4 ⊢ [𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) |
10 | sbim 2293 | . . . . 5 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) ↔ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → [𝑙 / 𝑘]𝐵 = 𝑍)) | |
11 | sban 2076 | . . . . . . 7 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) ↔ ([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ (𝐴 ∖ 𝑊))) | |
12 | suppss2f.p | . . . . . . . . 9 ⊢ Ⅎ𝑘𝜑 | |
13 | 12 | sbf 2258 | . . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝜑 ↔ 𝜑) |
14 | suppss2f.w | . . . . . . . . . 10 ⊢ Ⅎ𝑘𝑊 | |
15 | 1, 14 | nfdif 4121 | . . . . . . . . 9 ⊢ Ⅎ𝑘(𝐴 ∖ 𝑊) |
16 | 15 | clelsb1fw 2903 | . . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝑘 ∈ (𝐴 ∖ 𝑊) ↔ 𝑙 ∈ (𝐴 ∖ 𝑊)) |
17 | 13, 16 | anbi12i 627 | . . . . . . 7 ⊢ (([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ (𝐴 ∖ 𝑊)) ↔ (𝜑 ∧ 𝑙 ∈ (𝐴 ∖ 𝑊))) |
18 | 11, 17 | bitri 275 | . . . . . 6 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) ↔ (𝜑 ∧ 𝑙 ∈ (𝐴 ∖ 𝑊))) |
19 | sbsbc 3779 | . . . . . . 7 ⊢ ([𝑙 / 𝑘]𝐵 = 𝑍 ↔ [𝑙 / 𝑘]𝐵 = 𝑍) | |
20 | sbceq1g 4410 | . . . . . . . 8 ⊢ (𝑙 ∈ V → ([𝑙 / 𝑘]𝐵 = 𝑍 ↔ ⦋𝑙 / 𝑘⦌𝐵 = 𝑍)) | |
21 | 20 | elv 3476 | . . . . . . 7 ⊢ ([𝑙 / 𝑘]𝐵 = 𝑍 ↔ ⦋𝑙 / 𝑘⦌𝐵 = 𝑍) |
22 | 19, 21 | bitri 275 | . . . . . 6 ⊢ ([𝑙 / 𝑘]𝐵 = 𝑍 ↔ ⦋𝑙 / 𝑘⦌𝐵 = 𝑍) |
23 | 18, 22 | imbi12i 350 | . . . . 5 ⊢ (([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → [𝑙 / 𝑘]𝐵 = 𝑍) ↔ ((𝜑 ∧ 𝑙 ∈ (𝐴 ∖ 𝑊)) → ⦋𝑙 / 𝑘⦌𝐵 = 𝑍)) |
24 | 10, 23 | bitri 275 | . . . 4 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) ↔ ((𝜑 ∧ 𝑙 ∈ (𝐴 ∖ 𝑊)) → ⦋𝑙 / 𝑘⦌𝐵 = 𝑍)) |
25 | 9, 24 | mpbi 229 | . . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐴 ∖ 𝑊)) → ⦋𝑙 / 𝑘⦌𝐵 = 𝑍) |
26 | suppss2f.v | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
27 | 25, 26 | suppss2 8199 | . 2 ⊢ (𝜑 → ((𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵) supp 𝑍) ⊆ 𝑊) |
28 | 7, 27 | eqsstrid 4026 | 1 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 Ⅎwnf 1778 [wsb 2060 ∈ wcel 2099 Ⅎwnfc 2879 Vcvv 3470 [wsbc 3775 ⦋csb 3890 ∖ cdif 3942 ⊆ wss 3945 ↦ cmpt 5225 (class class class)co 7414 supp csupp 8159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-supp 8160 |
This theorem is referenced by: elrspunidl 33138 esumss 33685 |
Copyright terms: Public domain | W3C validator |