Theorem suppss2f 30047

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.)

Hypotheses

Ref	Expression
suppss2f.p	⊢ Ⅎ𝑘𝜑
suppss2f.a	⊢ Ⅎ𝑘𝐴
suppss2f.w	⊢ Ⅎ𝑘𝑊
suppss2f.n	⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍)
suppss2f.v	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
suppss2f	⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊)

Distinct variable group:   𝑘,𝑍

Allowed substitution hints:   𝜑(𝑘)   𝐴(𝑘)   𝐵(𝑘)   𝑉(𝑘)   𝑊(𝑘)

Proof of Theorem suppss2f

Dummy variable 𝑙 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suppss2f.a	. . . 4 ⊢ Ⅎ𝑘𝐴
2		nfcv 2947	. . . 4 ⊢ Ⅎ𝑙𝐴
3		nfcv 2947	. . . 4 ⊢ Ⅎ𝑙𝐵
4		nfcsb1v 3828	. . . 4 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐵
5		csbeq1a 3819	. . . 4 ⊢ (𝑘 = 𝑙 → 𝐵 = ⦋𝑙 / 𝑘⦌𝐵)
6	1, 2, 3, 4, 5	cbvmptf 5053	. . 3 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵)
7	6	oveq1i 7017	. 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 2042	. . . 4 ⊢ [𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍)
10		sbim 2275	. . . . 5 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) ↔ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → [𝑙 / 𝑘]𝐵 = 𝑍))
11		sban 2057	. . . . . . 7 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) ↔ ([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ (𝐴 ∖ 𝑊)))
12		suppss2f.p	. . . . . . . . 9 ⊢ Ⅎ𝑘𝜑
13	12	sbf 2232	. . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝜑 ↔ 𝜑)
14		suppss2f.w	. . . . . . . . . 10 ⊢ Ⅎ𝑘𝑊
15	1, 14	nfdif 4018	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝐴 ∖ 𝑊)
16	15	clelsb3f 2951	. . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝑘 ∈ (𝐴 ∖ 𝑊) ↔ 𝑙 ∈ (𝐴 ∖ 𝑊))
17	13, 16	anbi12i 626	. . . . . . 7 ⊢ (([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ (𝐴 ∖ 𝑊)) ↔ (𝜑 ∧ 𝑙 ∈ (𝐴 ∖ 𝑊)))
18	11, 17	bitri 276	. . . . . 6 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) ↔ (𝜑 ∧ 𝑙 ∈ (𝐴 ∖ 𝑊)))
19		sbsbc 3705	. . . . . . 7 ⊢ ([𝑙 / 𝑘]𝐵 = 𝑍 ↔ [𝑙 / 𝑘]𝐵 = 𝑍)
20		sbceq1g 4280	. . . . . . . 8 ⊢ (𝑙 ∈ V → ([𝑙 / 𝑘]𝐵 = 𝑍 ↔ ⦋𝑙 / 𝑘⦌𝐵 = 𝑍))
21	20	elv 3437	. . . . . . 7 ⊢ ([𝑙 / 𝑘]𝐵 = 𝑍 ↔ ⦋𝑙 / 𝑘⦌𝐵 = 𝑍)
22	19, 21	bitri 276	. . . . . 6 ⊢ ([𝑙 / 𝑘]𝐵 = 𝑍 ↔ ⦋𝑙 / 𝑘⦌𝐵 = 𝑍)
23	18, 22	imbi12i 352	. . . . 5 ⊢ (([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → [𝑙 / 𝑘]𝐵 = 𝑍) ↔ ((𝜑 ∧ 𝑙 ∈ (𝐴 ∖ 𝑊)) → ⦋𝑙 / 𝑘⦌𝐵 = 𝑍))
24	10, 23	bitri 276	. . . 4 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) ↔ ((𝜑 ∧ 𝑙 ∈ (𝐴 ∖ 𝑊)) → ⦋𝑙 / 𝑘⦌𝐵 = 𝑍))
25	9, 24	mpbi 231	. . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐴 ∖ 𝑊)) → ⦋𝑙 / 𝑘⦌𝐵 = 𝑍)
26		suppss2f.v	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
27	25, 26	suppss2 7706	. 2 ⊢ (𝜑 → ((𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵) supp 𝑍) ⊆ 𝑊)
28	7, 27	syl5eqss 3931	1 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1520  Ⅎwnf 1763  [wsb 2040   ∈ wcel 2079  Ⅎwnfc 2931  Vcvv 3432  [wsbc 3701  ⦋csb 3806   ∖ cdif 3851   ⊆ wss 3854   ↦ cmpt 5035  (class class class)co 7007   supp csupp 7672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-ov 7010  df-oprab 7011  df-mpo 7012  df-supp 7673

This theorem is referenced by:  esumss  30904