Theorem fdifsuppconst 32619

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 6549	. . . . . 6 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2	1	biimpi 216	. . . . 5 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹)
3	2	ad2antrr 726	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) → 𝐹 Fn dom 𝐹)
4		fdifsuppconst.1	. . . . 5 ⊢ 𝐴 = (dom 𝐹 ∖ (𝐹 supp 𝑍))
5		difssd 4103	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) ⊆ dom 𝐹)
6	4, 5	eqsstrid 3988	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) → 𝐴 ⊆ dom 𝐹)
7	3, 6	fnssresd 6645	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) → (𝐹 ↾ 𝐴) Fn 𝐴)
8		fnconstg 6751	. . . 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 7880	. . . . . 6 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
12	11	ad3antlr 731	. . . . 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 2821	. . . . . . 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 8153	. . . 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 6881	. . . 4 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
20		fvconst2g 7179	. . . . 5 ⊢ ((𝑍 ∈ 𝑊 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝑍})‘𝑥) = 𝑍)
21	20	adantll 714	. . . 4 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝑍})‘𝑥) = 𝑍)
22	17, 19, 21	3eqtr4d 2775	. . 3 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) = ((𝐴 × {𝑍})‘𝑥))
23	7, 9, 22	eqfnfvd 7009	. 2 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) → (𝐹 ↾ 𝐴) = (𝐴 × {𝑍}))
24	23	3impa 1109	1 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 ↾ 𝐴) = (𝐴 × {𝑍}))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ∖ cdif 3914  {csn 4592   × cxp 5639  dom cdm 5641   ↾ cres 5643  Fun wfun 6508   Fn wfn 6509  ‘cfv 6514  (class class class)co 7390   supp csupp 8142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-supp 8143

This theorem is referenced by: (None)