Theorem fdifsuppconst 32674

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 6516	. . . . . 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 4086	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) ⊆ dom 𝐹)
6	4, 5	eqsstrid 3969	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) → 𝐴 ⊆ dom 𝐹)
7	3, 6	fnssresd 6610	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) → (𝐹 ↾ 𝐴) Fn 𝐴)
8		fnconstg 6716	. . . 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 7837	. . . . . 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 2825	. . . . . . 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 8107	. . . 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 6848	. . . 4 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
20		fvconst2g 7142	. . . . 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 2778	. . 3 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) = ((𝐴 × {𝑍})‘𝑥))
23	7, 9, 22	eqfnfvd 6973	. 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 1541   ∈ wcel 2113  Vcvv 3437   ∖ cdif 3895  {csn 4575   × cxp 5617  dom cdm 5619   ↾ cres 5621  Fun wfun 6480   Fn wfn 6481  ‘cfv 6486  (class class class)co 7352   supp csupp 8096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-supp 8097

This theorem is referenced by: (None)