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 2751 | . . . 4 ⊢ Ⅎ𝑙𝐴 | |
3 | nfcv 2751 | . . . 4 ⊢ Ⅎ𝑙𝐵 | |
4 | nfcsb1v 3515 | . . . 4 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐵 | |
5 | csbeq1a 3508 | . . . 4 ⊢ (𝑘 = 𝑙 → 𝐵 = ⦋𝑙 / 𝑘⦌𝐵) | |
6 | 1, 2, 3, 4, 5 | cbvmptf 4676 | . . 3 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵) |
7 | 6 | oveq1i 6559 | . 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 2407 | . . . 4 ⊢ [𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) |
10 | sbim 2383 | . . . . 5 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) ↔ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → [𝑙 / 𝑘]𝐵 = 𝑍)) | |
11 | sban 2387 | . . . . . . 7 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) ↔ ([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ (𝐴 ∖ 𝑊))) | |
12 | suppss2f.p | . . . . . . . . 9 ⊢ Ⅎ𝑘𝜑 | |
13 | 12 | sbf 2368 | . . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝜑 ↔ 𝜑) |
14 | suppss2f.w | . . . . . . . . . 10 ⊢ Ⅎ𝑘𝑊 | |
15 | 1, 14 | nfdif 3693 | . . . . . . . . 9 ⊢ Ⅎ𝑘(𝐴 ∖ 𝑊) |
16 | 15 | clelsb3f 28704 | . . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝑘 ∈ (𝐴 ∖ 𝑊) ↔ 𝑙 ∈ (𝐴 ∖ 𝑊)) |
17 | 13, 16 | anbi12i 729 | . . . . . . 7 ⊢ (([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ (𝐴 ∖ 𝑊)) ↔ (𝜑 ∧ 𝑙 ∈ (𝐴 ∖ 𝑊))) |
18 | 11, 17 | bitri 263 | . . . . . 6 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) ↔ (𝜑 ∧ 𝑙 ∈ (𝐴 ∖ 𝑊))) |
19 | sbsbc 3406 | . . . . . . 7 ⊢ ([𝑙 / 𝑘]𝐵 = 𝑍 ↔ [𝑙 / 𝑘]𝐵 = 𝑍) | |
20 | vex 3176 | . . . . . . . 8 ⊢ 𝑙 ∈ V | |
21 | sbceq1g 3940 | . . . . . . . 8 ⊢ (𝑙 ∈ V → ([𝑙 / 𝑘]𝐵 = 𝑍 ↔ ⦋𝑙 / 𝑘⦌𝐵 = 𝑍)) | |
22 | 20, 21 | ax-mp 5 | . . . . . . 7 ⊢ ([𝑙 / 𝑘]𝐵 = 𝑍 ↔ ⦋𝑙 / 𝑘⦌𝐵 = 𝑍) |
23 | 19, 22 | bitri 263 | . . . . . 6 ⊢ ([𝑙 / 𝑘]𝐵 = 𝑍 ↔ ⦋𝑙 / 𝑘⦌𝐵 = 𝑍) |
24 | 18, 23 | imbi12i 339 | . . . . 5 ⊢ (([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → [𝑙 / 𝑘]𝐵 = 𝑍) ↔ ((𝜑 ∧ 𝑙 ∈ (𝐴 ∖ 𝑊)) → ⦋𝑙 / 𝑘⦌𝐵 = 𝑍)) |
25 | 10, 24 | bitri 263 | . . . 4 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) ↔ ((𝜑 ∧ 𝑙 ∈ (𝐴 ∖ 𝑊)) → ⦋𝑙 / 𝑘⦌𝐵 = 𝑍)) |
26 | 9, 25 | mpbi 219 | . . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐴 ∖ 𝑊)) → ⦋𝑙 / 𝑘⦌𝐵 = 𝑍) |
27 | suppss2f.v | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
28 | 26, 27 | suppss2 7216 | . 2 ⊢ (𝜑 → ((𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵) supp 𝑍) ⊆ 𝑊) |
29 | 7, 28 | syl5eqss 3612 | 1 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 [wsb 1867 ∈ wcel 1977 Ⅎwnfc 2738 Vcvv 3173 [wsbc 3402 ⦋csb 3499 ∖ cdif 3537 ⊆ wss 3540 ↦ cmpt 4643 (class class class)co 6549 supp csupp 7182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-supp 7183 |
This theorem is referenced by: esumss 29461 |
Copyright terms: Public domain | W3C validator |