fvdifsupp - Mathbox for Thierry Arnoux

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Theorem fvdifsupp 32342

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 3953	. 2 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝐹 supp 𝑍))
3	1	eldifad 3952	. . . 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 8150	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍)))
8	4, 5, 6, 7	syl3anc 1368	. . . 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 2976	. 2 ⊢ (𝜑 → ((𝐹‘𝑋) = 𝑍 ↔ ¬ 𝑋 ∈ (𝐹 supp 𝑍)))
11	2, 10	mpbird 257	1 ⊢ (𝜑 → (𝐹‘𝑋) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∖ cdif 3937 Fn wfn 6528 ‘cfv 6533 (class class class)co 7401 supp csupp 8140

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-supp 8141

This theorem is referenced by: fdifsuppconst 32346 elrspunidl 32981 elrspunsn 32982 evls1fpws 33079

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