Theorem suppssrst 6495

Description: A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.)

Hypotheses

Ref	Expression
suppssr.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
suppssr.n	⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)
suppssr.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
suppssrst.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
suppssrst.st	⊢ (𝜑 → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 STAB 𝑢 = 𝑣)

Assertion

Ref	Expression
suppssrst	⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍)

Distinct variable groups:   𝑢,𝐵,𝑣   𝑢,𝐹,𝑣   𝑢,𝑋,𝑣   𝑣,𝑍

Allowed substitution hints:   𝜑(𝑣,𝑢)   𝐴(𝑣,𝑢)   𝑉(𝑣,𝑢)   𝑊(𝑣,𝑢)   𝑍(𝑢)

Proof of Theorem suppssrst

Step	Hyp	Ref	Expression
1		eldif 3229	. 2 ⊢ (𝑋 ∈ (𝐴 ∖ 𝑊) ↔ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑊))
2		df-ne 2421	. . . . . 6 ⊢ ((𝐹‘𝑋) ≠ 𝑍 ↔ ¬ (𝐹‘𝑋) = 𝑍)
3		suppssr.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
4		suppssr.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5	3, 4	fexd 5942	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ V)
6		fvexg 5712	. . . . . . . . . 10 ⊢ ((𝐹 ∈ V ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ V)
7	5, 6	sylan 283	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ V)
8	7	biantrurd 305	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ≠ 𝑍 ↔ ((𝐹‘𝑋) ∈ V ∧ (𝐹‘𝑋) ≠ 𝑍)))
9		eldifsn 3839	. . . . . . . 8 ⊢ ((𝐹‘𝑋) ∈ (V ∖ {𝑍}) ↔ ((𝐹‘𝑋) ∈ V ∧ (𝐹‘𝑋) ≠ 𝑍))
10	8, 9	bitr4di 198	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ≠ 𝑍 ↔ (𝐹‘𝑋) ∈ (V ∖ {𝑍})))
11	3	ffnd 5532	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn 𝐴)
12		suppssrst.z	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
13		elsuppfn 6477	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝐵) → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍)))
14	11, 4, 12, 13	syl3anc 1278	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍)))
15	10	pm5.32da 456	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ∈ (V ∖ {𝑍}))))
16	14, 15	bitrd 188	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ∈ (V ∖ {𝑍}))))
17		suppssr.n	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)
18	17	sseld 3247	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) → 𝑋 ∈ 𝑊))
19	16, 18	sylbird 170	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ∈ (V ∖ {𝑍})) → 𝑋 ∈ 𝑊))
20	19	expdimp 259	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ∈ (V ∖ {𝑍}) → 𝑋 ∈ 𝑊))
21	10, 20	sylbid 150	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ≠ 𝑍 → 𝑋 ∈ 𝑊))
22	2, 21	biimtrrid 153	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (¬ (𝐹‘𝑋) = 𝑍 → 𝑋 ∈ 𝑊))
23	22	con3d 640	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (¬ 𝑋 ∈ 𝑊 → ¬ ¬ (𝐹‘𝑋) = 𝑍))
24		eqeq2 2248	. . . . . . 7 ⊢ (𝑣 = 𝑍 → ((𝐹‘𝑋) = 𝑣 ↔ (𝐹‘𝑋) = 𝑍))
25	24	stbid 844	. . . . . 6 ⊢ (𝑣 = 𝑍 → (STAB (𝐹‘𝑋) = 𝑣 ↔ STAB (𝐹‘𝑋) = 𝑍))
26		eqeq1 2245	. . . . . . . . 9 ⊢ (𝑢 = (𝐹‘𝑋) → (𝑢 = 𝑣 ↔ (𝐹‘𝑋) = 𝑣))
27	26	stbid 844	. . . . . . . 8 ⊢ (𝑢 = (𝐹‘𝑋) → (STAB 𝑢 = 𝑣 ↔ STAB (𝐹‘𝑋) = 𝑣))
28	27	ralbidv 2550	. . . . . . 7 ⊢ (𝑢 = (𝐹‘𝑋) → (∀𝑣 ∈ 𝐵 STAB 𝑢 = 𝑣 ↔ ∀𝑣 ∈ 𝐵 STAB (𝐹‘𝑋) = 𝑣))
29		suppssrst.st	. . . . . . . 8 ⊢ (𝜑 → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 STAB 𝑢 = 𝑣)
30	29	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 STAB 𝑢 = 𝑣)
31	3	ffvelcdmda 5837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ 𝐵)
32	28, 30, 31	rspcdva 2934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ∀𝑣 ∈ 𝐵 STAB (𝐹‘𝑋) = 𝑣)
33	12	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑍 ∈ 𝐵)
34	25, 32, 33	rspcdva 2934	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → STAB (𝐹‘𝑋) = 𝑍)
35		df-stab 843	. . . . 5 ⊢ (STAB (𝐹‘𝑋) = 𝑍 ↔ (¬ ¬ (𝐹‘𝑋) = 𝑍 → (𝐹‘𝑋) = 𝑍))
36	34, 35	sylib 122	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (¬ ¬ (𝐹‘𝑋) = 𝑍 → (𝐹‘𝑋) = 𝑍))
37	23, 36	syld 45	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (¬ 𝑋 ∈ 𝑊 → (𝐹‘𝑋) = 𝑍))
38	37	impr 379	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑊)) → (𝐹‘𝑋) = 𝑍)
39	1, 38	sylan2b 287	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  STAB wstab 842   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  ∀wral 2528  Vcvv 2821   ∖ cdif 3217   ⊆ wss 3220  {csn 3708   Fn wfn 5370  ⟶wf 5371  ‘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-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-stab 843  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-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  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-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-supp 6470

This theorem is referenced by: (None)