Theorem suppss2f 32711

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 2898	. . . 4 ⊢ Ⅎ𝑙𝐴
3		nfcv 2898	. . . 4 ⊢ Ⅎ𝑙𝐵
4		nfcsb1v 3861	. . . 4 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐵
5		csbeq1a 3851	. . . 4 ⊢ (𝑘 = 𝑙 → 𝐵 = ⦋𝑙 / 𝑘⦌𝐵)
6	1, 2, 3, 4, 5	cbvmptf 5185	. . 3 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵)
7	6	oveq1i 7377	. 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 2072	. . . 4 ⊢ [𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍)
10		sbim 2310	. . . . 5 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍) ↔ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → [𝑙 / 𝑘]𝐵 = 𝑍))
11		sban 2086	. . . . . . 7 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) ↔ ([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ (𝐴 ∖ 𝑊)))
12		suppss2f.p	. . . . . . . . 9 ⊢ Ⅎ𝑘𝜑
13	12	sbf 2278	. . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝜑 ↔ 𝜑)
14		suppss2f.w	. . . . . . . . . 10 ⊢ Ⅎ𝑘𝑊
15	1, 14	nfdif 4069	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝐴 ∖ 𝑊)
16	15	clelsb1fw 2902	. . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝑘 ∈ (𝐴 ∖ 𝑊) ↔ 𝑙 ∈ (𝐴 ∖ 𝑊))
17	13, 16	anbi12i 629	. . . . . . 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 3732	. . . . . . 7 ⊢ ([𝑙 / 𝑘]𝐵 = 𝑍 ↔ [𝑙 / 𝑘]𝐵 = 𝑍)
20		sbceq1g 4357	. . . . . . . 8 ⊢ (𝑙 ∈ V → ([𝑙 / 𝑘]𝐵 = 𝑍 ↔ ⦋𝑙 / 𝑘⦌𝐵 = 𝑍))
21	20	elv 3434	. . . . . . 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 8150	. 2 ⊢ (𝜑 → ((𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵) supp 𝑍) ⊆ 𝑊)
28	7, 27	eqsstrid 3960	1 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  Ⅎwnf 1785  [wsb 2068   ∈ wcel 2114  Ⅎwnfc 2883  Vcvv 3429  [wsbc 3728  ⦋csb 3837   ∖ cdif 3886   ⊆ wss 3889   ↦ cmpt 5166  (class class class)co 7367   supp csupp 8110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-supp 8111

This theorem is referenced by:  suppgsumssiun  33133  elrspunidl  33488  esumss  34216