Theorem fczsupp0 6493

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 6470	. . . . . 6 ⊢ supp = (𝑓 ∈ V, 𝑧 ∈ V ↦ {𝑞 ∈ dom 𝑓 ∣ (𝑓 “ {𝑞}) ≠ {𝑧}})
2	1	elmpocl 6278	. . . . 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 5588	. . . . . . . 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 6476	. . . . . . 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 1278	. . . . . 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 5923	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝑥 ∈ 𝐵) → ((𝐵 × {𝑍})‘𝑥) = 𝑍)
12	3, 10, 11	syl2anc 415	. . 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 2441	. . 3 ⊢ (𝑥 ∈ ((𝐵 × {𝑍}) supp 𝑍) → ¬ ((𝐵 × {𝑍})‘𝑥) = 𝑍)
15	12, 14	pm2.65i 648	. 2 ⊢ ¬ 𝑥 ∈ ((𝐵 × {𝑍}) supp 𝑍)
16	15	nel0 3543	1 ⊢ ((𝐵 × {𝑍}) supp 𝑍) = ∅

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  {crab 2532  Vcvv 2821  ∅c0 3520  {csn 3708   × cxp 4770  dom cdm 4772   “ cima 4775   Fn wfn 5370  ‘cfv 5375  (class class class)co 6079   supp csupp 6469

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-supp 6470

This theorem is referenced by:  fczfsuppd  7291