fvdifsupp - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fvdifsupp		Structured version Visualization version GIF version

Theorem fvdifsupp 31018

Description: Function value is zero outside of its support. (Contributed by Thierry Arnoux, 21-Jan-2024.)

**Hypotheses**
Ref	Expression
fvdifsupp.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
fvdifsupp.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fvdifsupp.3	⊢ (𝜑 → 𝑍 ∈ 𝑊)
fvdifsupp.4	⊢ (𝜑 → 𝑋 ∈ (𝐴 ∖ (𝐹 supp 𝑍)))

**Assertion**
Ref	Expression
fvdifsupp	⊢ (𝜑 → (𝐹‘𝑋) = 𝑍)

**Proof of Theorem fvdifsupp**
Step	Hyp	Ref	Expression
1		fvdifsupp.4	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∖ (𝐹 supp 𝑍)))
2	1	eldifbd 3900	. 2 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝐹 supp 𝑍))
3	1	eldifad 3899	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
4		fvdifsupp.1	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐴)
5		fvdifsupp.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		fvdifsupp.3	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
7		elsuppfn 7987	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍)))
8	4, 5, 6, 7	syl3anc 1370	. . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍)))
9	3, 8	mpbirand 704	. . 3 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝐹‘𝑋) ≠ 𝑍))
10	9	necon2bbid 2987	. 2 ⊢ (𝜑 → ((𝐹‘𝑋) = 𝑍 ↔ ¬ 𝑋 ∈ (𝐹 supp 𝑍)))
11	2, 10	mpbird 256	1 ⊢ (𝜑 → (𝐹‘𝑋) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3884 Fn wfn 6428 ‘cfv 6433 (class class class)co 7275 supp csupp 7977

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-supp 7978

This theorem is referenced by: fdifsuppconst 31023 elrspunidl 31606

W3C validator