Theorem fdifsuppconst 32701

Description: A function is a zero constant outside of its support. (Contributed by Thierry Arnoux, 22-Jun-2024.)

Hypothesis

Ref	Expression
fdifsuppconst.1	⊢ 𝐴 = (dom 𝐹 ∖ (𝐹 supp 𝑍))

Assertion

Ref	Expression
fdifsuppconst	⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 ↾ 𝐴) = (𝐴 × {𝑍}))

Proof of Theorem fdifsuppconst

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfn 6608	. . . . . 6 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2	1	biimpi 216	. . . . 5 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹)
3	2	ad2antrr 725	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) → 𝐹 Fn dom 𝐹)
4		fdifsuppconst.1	. . . . 5 ⊢ 𝐴 = (dom 𝐹 ∖ (𝐹 supp 𝑍))
5		difssd 4160	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) ⊆ dom 𝐹)
6	4, 5	eqsstrid 4057	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) → 𝐴 ⊆ dom 𝐹)
7	3, 6	fnssresd 6704	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) → (𝐹 ↾ 𝐴) Fn 𝐴)
8		fnconstg 6809	. . . 4 ⊢ (𝑍 ∈ 𝑊 → (𝐴 × {𝑍}) Fn 𝐴)
9	8	adantl 481	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) → (𝐴 × {𝑍}) Fn 𝐴)
10	3	adantr 480	. . . . 5 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → 𝐹 Fn dom 𝐹)
11		dmexg 7941	. . . . . 6 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
12	11	ad3antlr 730	. . . . 5 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → dom 𝐹 ∈ V)
13		simplr 768	. . . . 5 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → 𝑍 ∈ 𝑊)
14	4	eleq2i 2836	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)))
15	14	biimpi 216	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)))
16	15	adantl 481	. . . . 5 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)))
17	10, 12, 13, 16	fvdifsupp 8212	. . . 4 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝑍)
18		simpr 484	. . . . 5 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
19	18	fvresd 6940	. . . 4 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
20		fvconst2g 7239	. . . . 5 ⊢ ((𝑍 ∈ 𝑊 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝑍})‘𝑥) = 𝑍)
21	20	adantll 713	. . . 4 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝑍})‘𝑥) = 𝑍)
22	17, 19, 21	3eqtr4d 2790	. . 3 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) = ((𝐴 × {𝑍})‘𝑥))
23	7, 9, 22	eqfnfvd 7067	. 2 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) → (𝐹 ↾ 𝐴) = (𝐴 × {𝑍}))
24	23	3impa 1110	1 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 ↾ 𝐴) = (𝐴 × {𝑍}))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∖ cdif 3973  {csn 4648   × cxp 5698  dom cdm 5700   ↾ cres 5702  Fun wfun 6567   Fn wfn 6568  ‘cfv 6573  (class class class)co 7448   supp csupp 8201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-supp 8202

This theorem is referenced by: (None)