Theorem suppssrgst 6475

Description: A function is zero outside its support. Version of suppssrst 6474 avoiding ax-coll 4230 by assuming 𝐹 is a set rather than its domain 𝐴. (Contributed by SN, 5-May-2024.)

Hypotheses

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

Assertion

Ref	Expression
suppssrgst	⊢ ((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 suppssrgst

Step	Hyp	Ref	Expression
1		eldif 3223	. 2 ⊢ (𝑋 ∈ (𝐴 ∖ 𝑊) ↔ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑊))
2		df-ne 2415	. . . . . 6 ⊢ ((𝐹‘𝑋) ≠ 𝑍 ↔ ¬ (𝐹‘𝑋) = 𝑍)
3		suppssrg.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
4		fvexg 5694	. . . . . . . . . 10 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ V)
5	3, 4	sylan 283	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ V)
6	5	biantrurd 305	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ≠ 𝑍 ↔ ((𝐹‘𝑋) ∈ V ∧ (𝐹‘𝑋) ≠ 𝑍)))
7		eldifsn 3825	. . . . . . . 8 ⊢ ((𝐹‘𝑋) ∈ (V ∖ {𝑍}) ↔ ((𝐹‘𝑋) ∈ V ∧ (𝐹‘𝑋) ≠ 𝑍))
8	6, 7	bitr4di 198	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ≠ 𝑍 ↔ (𝐹‘𝑋) ∈ (V ∖ {𝑍})))
9		suppssrg.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
10	9	ffnd 5514	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn 𝐴)
11		suppssrgst.z	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
12		elsuppfng 6455	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝐵) → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍)))
13	10, 3, 11, 12	syl3anc 1274	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍)))
14	8	pm5.32da 452	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ∈ (V ∖ {𝑍}))))
15	13, 14	bitrd 188	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ∈ (V ∖ {𝑍}))))
16		suppssrg.n	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)
17	16	sseld 3241	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) → 𝑋 ∈ 𝑊))
18	15, 17	sylbird 170	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ∈ (V ∖ {𝑍})) → 𝑋 ∈ 𝑊))
19	18	expdimp 259	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ∈ (V ∖ {𝑍}) → 𝑋 ∈ 𝑊))
20	8, 19	sylbid 150	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ≠ 𝑍 → 𝑋 ∈ 𝑊))
21	2, 20	biimtrrid 153	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (¬ (𝐹‘𝑋) = 𝑍 → 𝑋 ∈ 𝑊))
22	21	con3d 636	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (¬ 𝑋 ∈ 𝑊 → ¬ ¬ (𝐹‘𝑋) = 𝑍))
23		eqeq2 2244	. . . . . . 7 ⊢ (𝑣 = 𝑍 → ((𝐹‘𝑋) = 𝑣 ↔ (𝐹‘𝑋) = 𝑍))
24	23	stbid 840	. . . . . 6 ⊢ (𝑣 = 𝑍 → (STAB (𝐹‘𝑋) = 𝑣 ↔ STAB (𝐹‘𝑋) = 𝑍))
25		eqeq1 2241	. . . . . . . . 9 ⊢ (𝑢 = (𝐹‘𝑋) → (𝑢 = 𝑣 ↔ (𝐹‘𝑋) = 𝑣))
26	25	stbid 840	. . . . . . . 8 ⊢ (𝑢 = (𝐹‘𝑋) → (STAB 𝑢 = 𝑣 ↔ STAB (𝐹‘𝑋) = 𝑣))
27	26	ralbidv 2544	. . . . . . 7 ⊢ (𝑢 = (𝐹‘𝑋) → (∀𝑣 ∈ 𝐵 STAB 𝑢 = 𝑣 ↔ ∀𝑣 ∈ 𝐵 STAB (𝐹‘𝑋) = 𝑣))
28		suppssrgst.st	. . . . . . . 8 ⊢ (𝜑 → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 STAB 𝑢 = 𝑣)
29	28	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 STAB 𝑢 = 𝑣)
30	9	ffvelcdmda 5817	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ 𝐵)
31	27, 29, 30	rspcdva 2928	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ∀𝑣 ∈ 𝐵 STAB (𝐹‘𝑋) = 𝑣)
32	11	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑍 ∈ 𝐵)
33	24, 31, 32	rspcdva 2928	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → STAB (𝐹‘𝑋) = 𝑍)
34		df-stab 839	. . . . 5 ⊢ (STAB (𝐹‘𝑋) = 𝑍 ↔ (¬ ¬ (𝐹‘𝑋) = 𝑍 → (𝐹‘𝑋) = 𝑍))
35	33, 34	sylib 122	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (¬ ¬ (𝐹‘𝑋) = 𝑍 → (𝐹‘𝑋) = 𝑍))
36	22, 35	syld 45	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (¬ 𝑋 ∈ 𝑊 → (𝐹‘𝑋) = 𝑍))
37	36	impr 379	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑊)) → (𝐹‘𝑋) = 𝑍)
38	1, 37	sylan2b 287	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  STAB wstab 838   = wceq 1398   ∈ wcel 2205   ≠ wne 2414  ∀wral 2522  Vcvv 2815   ∖ cdif 3211   ⊆ wss 3214  {csn 3694   Fn wfn 5352  ⟶wf 5353  ‘cfv 5357  (class class class)co 6058   supp csupp 6448

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-stab 839  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-supp 6449

This theorem is referenced by: (None)