Theorem fczsupp0 6437

Description: The support of a constant function with value zero is empty. (Contributed by AV, 30-Jun-2019.)

Assertion

Ref	Expression
fczsupp0	⊢ ((𝐵 × {𝑍}) supp 𝑍) = ∅

Proof of Theorem fczsupp0

Dummy variables 𝑥 𝑓 𝑞 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-supp 6414	. . . . . 6 ⊢ supp = (𝑓 ∈ V, 𝑧 ∈ V ↦ {𝑞 ∈ dom 𝑓 ∣ (𝑓 “ {𝑞}) ≠ {𝑧}})
2	1	elmpocl 6227	. . . . 5 ⊢ (𝑥 ∈ ((𝐵 × {𝑍}) supp 𝑍) → ((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V))
3	2	simprd 114	. . . 4 ⊢ (𝑥 ∈ ((𝐵 × {𝑍}) supp 𝑍) → 𝑍 ∈ V)
4		fnconstg 5543	. . . . . . . 8 ⊢ (𝑍 ∈ V → (𝐵 × {𝑍}) Fn 𝐵)
5	3, 4	syl 14	. . . . . . 7 ⊢ (𝑥 ∈ ((𝐵 × {𝑍}) supp 𝑍) → (𝐵 × {𝑍}) Fn 𝐵)
6	2	simpld 112	. . . . . . 7 ⊢ (𝑥 ∈ ((𝐵 × {𝑍}) supp 𝑍) → (𝐵 × {𝑍}) ∈ V)
7		elsuppfng 6420	. . . . . . 7 ⊢ (((𝐵 × {𝑍}) Fn 𝐵 ∧ (𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → (𝑥 ∈ ((𝐵 × {𝑍}) supp 𝑍) ↔ (𝑥 ∈ 𝐵 ∧ ((𝐵 × {𝑍})‘𝑥) ≠ 𝑍)))
8	5, 6, 3, 7	syl3anc 1274	. . . . . 6 ⊢ (𝑥 ∈ ((𝐵 × {𝑍}) supp 𝑍) → (𝑥 ∈ ((𝐵 × {𝑍}) supp 𝑍) ↔ (𝑥 ∈ 𝐵 ∧ ((𝐵 × {𝑍})‘𝑥) ≠ 𝑍)))
9	8	ibi 176	. . . . 5 ⊢ (𝑥 ∈ ((𝐵 × {𝑍}) supp 𝑍) → (𝑥 ∈ 𝐵 ∧ ((𝐵 × {𝑍})‘𝑥) ≠ 𝑍))
10	9	simpld 112	. . . 4 ⊢ (𝑥 ∈ ((𝐵 × {𝑍}) supp 𝑍) → 𝑥 ∈ 𝐵)
11		fvconst2g 5876	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝑥 ∈ 𝐵) → ((𝐵 × {𝑍})‘𝑥) = 𝑍)
12	3, 10, 11	syl2anc 411	. . 3 ⊢ (𝑥 ∈ ((𝐵 × {𝑍}) supp 𝑍) → ((𝐵 × {𝑍})‘𝑥) = 𝑍)
13	9	simprd 114	. . . 4 ⊢ (𝑥 ∈ ((𝐵 × {𝑍}) supp 𝑍) → ((𝐵 × {𝑍})‘𝑥) ≠ 𝑍)
14	13	neneqd 2424	. . 3 ⊢ (𝑥 ∈ ((𝐵 × {𝑍}) supp 𝑍) → ¬ ((𝐵 × {𝑍})‘𝑥) = 𝑍)
15	12, 14	pm2.65i 644	. 2 ⊢ ¬ 𝑥 ∈ ((𝐵 × {𝑍}) supp 𝑍)
16	15	nel0 3518	1 ⊢ ((𝐵 × {𝑍}) supp 𝑍) = ∅

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202   ≠ wne 2403  {crab 2515  Vcvv 2803  ∅c0 3496  {csn 3673   × cxp 4729  dom cdm 4731   “ cima 4734   Fn wfn 5328  ‘cfv 5333  (class class class)co 6028   supp csupp 6413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-supp 6414

This theorem is referenced by: (None)