Theorem suppssrg 8227

Description: A function is zero outside its support. Version of suppssr 8226 avoiding ax-rep 5235 by assuming 𝐹 is a set rather than its domain 𝐴. (Contributed by SN, 5-May-2024.)

Hypotheses

Ref	Expression
suppssrg.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
suppssrg.n	⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)
suppssrg.a	⊢ (𝜑 → 𝐹 ∈ 𝑉)
suppssrg.z	⊢ (𝜑 → 𝑍 ∈ 𝑈)

Assertion

Ref	Expression
suppssrg	⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍)

Proof of Theorem suppssrg

Step	Hyp	Ref	Expression
1		eldif 4052	. 2 ⊢ (𝑋 ∈ (𝐴 ∖ 𝑊) ↔ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑊))
2		suppssrg.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
3	2	ffnd 6721	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
4		suppssrg.a	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
5		suppssrg.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
6		elsuppfng 8200	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑈) → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍)))
7	3, 4, 5, 6	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍)))
8		suppssrg.n	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)
9	8	sseld 3912	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) → 𝑋 ∈ 𝑊))
10	7, 9	sylbird 263	. . . . 5 ⊢ (𝜑 → ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍) → 𝑋 ∈ 𝑊))
11	10	expdimp 458	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ≠ 𝑍 → 𝑋 ∈ 𝑊))
12	11	necon1bd 2977	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (¬ 𝑋 ∈ 𝑊 → (𝐹‘𝑋) = 𝑍))
13	12	impr 460	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑊)) → (𝐹‘𝑋) = 𝑍)
14	1, 13	sylan2b 606	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2146   ≠ wne 2959   ⊆ wss 3900   ∖ cdif 4048   Fn wfn 6542  ⟶wf 6543  ‘cfv 6547  (class class class)co 7430   supp csupp 8190

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-fv 6555  df-ov 7433  df-oprab 7434  df-mpo 7435  df-supp 8191

This theorem is used by:  psrbaglesupp  22245  psrbaglefi  22249  evlsvvvallem2  22416  selvvvval  22466  mhpmulcl  22485  mhpvscacl  22490