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| 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 2905 | . . . 4 ⊢ Ⅎ𝑙𝐴 | |
| 3 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑙𝐵 | |
| 4 | nfcsb1v 3923 | . . . 4 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐵 | |
| 5 | csbeq1a 3913 | . . . 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 7441 | . 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 2066 | . . . 4 ⊢ [𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) |
| 10 | sbim 2303 | . . . . 5 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) ↔ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → [𝑙 / 𝑘]𝐵 = 𝑍)) | |
| 11 | sban 2080 | . . . . . . 7 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) ↔ ([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ (𝐴 ∖ 𝑊))) | |
| 12 | suppss2f.p | . . . . . . . . 9 ⊢ Ⅎ𝑘𝜑 | |
| 13 | 12 | sbf 2271 | . . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝜑 ↔ 𝜑) |
| 14 | suppss2f.w | . . . . . . . . . 10 ⊢ Ⅎ𝑘𝑊 | |
| 15 | 1, 14 | nfdif 4129 | . . . . . . . . 9 ⊢ Ⅎ𝑘(𝐴 ∖ 𝑊) |
| 16 | 15 | clelsb1fw 2909 | . . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝑘 ∈ (𝐴 ∖ 𝑊) ↔ 𝑙 ∈ (𝐴 ∖ 𝑊)) |
| 17 | 13, 16 | anbi12i 628 | . . . . . . 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 3792 | . . . . . . 7 ⊢ ([𝑙 / 𝑘]𝐵 = 𝑍 ↔ [𝑙 / 𝑘]𝐵 = 𝑍) | |
| 20 | sbceq1g 4417 | . . . . . . . 8 ⊢ (𝑙 ∈ V → ([𝑙 / 𝑘]𝐵 = 𝑍 ↔ ⦋𝑙 / 𝑘⦌𝐵 = 𝑍)) | |
| 21 | 20 | elv 3485 | . . . . . . 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 230 | . . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐴 ∖ 𝑊)) → ⦋𝑙 / 𝑘⦌𝐵 = 𝑍) |
| 26 | suppss2f.v | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 27 | 25, 26 | suppss2 8225 | . 2 ⊢ (𝜑 → ((𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵) supp 𝑍) ⊆ 𝑊) |
| 28 | 7, 27 | eqsstrid 4022 | 1 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 [wsb 2064 ∈ wcel 2108 Ⅎwnfc 2890 Vcvv 3480 [wsbc 3788 ⦋csb 3899 ∖ cdif 3948 ⊆ wss 3951 ↦ cmpt 5225 (class class class)co 7431 supp csupp 8185 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-supp 8186 |
| This theorem is referenced by: elrspunidl 33456 esumss 34073 |
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