Theorem suppss2f 29439

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 2764	. . . 4 ⊢ Ⅎ𝑙𝐴
3		nfcv 2764	. . . 4 ⊢ Ⅎ𝑙𝐵
4		nfcsb1v 3549	. . . 4 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐵
5		csbeq1a 3542	. . . 4 ⊢ (𝑘 = 𝑙 → 𝐵 = ⦋𝑙 / 𝑘⦌𝐵)
6	1, 2, 3, 4, 5	cbvmptf 4748	. . 3 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵)
7	6	oveq1i 6660	. 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 2419	. . . 4 ⊢ [𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍)
10		sbim 2395	. . . . 5 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) ↔ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → [𝑙 / 𝑘]𝐵 = 𝑍))
11		sban 2399	. . . . . . 7 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) ↔ ([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ (𝐴 ∖ 𝑊)))
12		suppss2f.p	. . . . . . . . 9 ⊢ Ⅎ𝑘𝜑
13	12	sbf 2380	. . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝜑 ↔ 𝜑)
14		suppss2f.w	. . . . . . . . . 10 ⊢ Ⅎ𝑘𝑊
15	1, 14	nfdif 3731	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝐴 ∖ 𝑊)
16	15	clelsb3f 2768	. . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝑘 ∈ (𝐴 ∖ 𝑊) ↔ 𝑙 ∈ (𝐴 ∖ 𝑊))
17	13, 16	anbi12i 733	. . . . . . 7 ⊢ (([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ (𝐴 ∖ 𝑊)) ↔ (𝜑 ∧ 𝑙 ∈ (𝐴 ∖ 𝑊)))
18	11, 17	bitri 264	. . . . . 6 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) ↔ (𝜑 ∧ 𝑙 ∈ (𝐴 ∖ 𝑊)))
19		sbsbc 3439	. . . . . . 7 ⊢ ([𝑙 / 𝑘]𝐵 = 𝑍 ↔ [𝑙 / 𝑘]𝐵 = 𝑍)
20		vex 3203	. . . . . . . 8 ⊢ 𝑙 ∈ V
21		sbceq1g 3988	. . . . . . . 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 264	. . . . . 6 ⊢ ([𝑙 / 𝑘]𝐵 = 𝑍 ↔ ⦋𝑙 / 𝑘⦌𝐵 = 𝑍)
24	18, 23	imbi12i 340	. . . . 5 ⊢ (([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → [𝑙 / 𝑘]𝐵 = 𝑍) ↔ ((𝜑 ∧ 𝑙 ∈ (𝐴 ∖ 𝑊)) → ⦋𝑙 / 𝑘⦌𝐵 = 𝑍))
25	10, 24	bitri 264	. . . 4 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) ↔ ((𝜑 ∧ 𝑙 ∈ (𝐴 ∖ 𝑊)) → ⦋𝑙 / 𝑘⦌𝐵 = 𝑍))
26	9, 25	mpbi 220	. . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐴 ∖ 𝑊)) → ⦋𝑙 / 𝑘⦌𝐵 = 𝑍)
27		suppss2f.v	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
28	26, 27	suppss2 7329	. 2 ⊢ (𝜑 → ((𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵) supp 𝑍) ⊆ 𝑊)
29	7, 28	syl5eqss 3649	1 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  Ⅎwnf 1708  [wsb 1880   ∈ wcel 1990  Ⅎwnfc 2751  Vcvv 3200  [wsbc 3435  ⦋csb 3533   ∖ cdif 3571   ⊆ wss 3574   ↦ cmpt 4729  (class class class)co 6650   supp csupp 7295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-supp 7296

This theorem is referenced by:  esumss  30134