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Theorem fczsupp0 7655
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
StepHypRef Expression
1 eqidd 2773 . . 3 (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → (𝐵 × {𝑍}) = (𝐵 × {𝑍}))
2 fnconstg 6390 . . . . 5 (𝑍 ∈ V → (𝐵 × {𝑍}) Fn 𝐵)
32adantl 474 . . . 4 (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → (𝐵 × {𝑍}) Fn 𝐵)
4 snnzg 4578 . . . . 5 (𝑍 ∈ V → {𝑍} ≠ ∅)
5 simpl 475 . . . . 5 (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → (𝐵 × {𝑍}) ∈ V)
6 xpexcnv 7434 . . . . 5 (({𝑍} ≠ ∅ ∧ (𝐵 × {𝑍}) ∈ V) → 𝐵 ∈ V)
74, 5, 6syl2an2 673 . . . 4 (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → 𝐵 ∈ V)
8 simpr 477 . . . 4 (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
9 fnsuppeq0 7654 . . . 4 (((𝐵 × {𝑍}) Fn 𝐵𝐵 ∈ V ∧ 𝑍 ∈ V) → (((𝐵 × {𝑍}) supp 𝑍) = ∅ ↔ (𝐵 × {𝑍}) = (𝐵 × {𝑍})))
103, 7, 8, 9syl3anc 1351 . . 3 (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → (((𝐵 × {𝑍}) supp 𝑍) = ∅ ↔ (𝐵 × {𝑍}) = (𝐵 × {𝑍})))
111, 10mpbird 249 . 2 (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → ((𝐵 × {𝑍}) supp 𝑍) = ∅)
12 supp0prc 7629 . 2 (¬ ((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → ((𝐵 × {𝑍}) supp 𝑍) = ∅)
1311, 12pm2.61i 177 1 ((𝐵 × {𝑍}) supp 𝑍) = ∅
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387   = wceq 1507  wcel 2048  wne 2961  Vcvv 3409  c0 4173  {csn 4435   × cxp 5398   Fn wfn 6177  (class class class)co 6970   supp csupp 7626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-supp 7627
This theorem is referenced by:  fczfsuppd  8638  cantnf  8942  mhp0cl  20033
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