Theorem suppssrg 8140

Description: A function is zero outside its support. Version of suppssr 8139 avoiding ax-rep 5213 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 3900	. 2 ⊢ (𝑋 ∈ (𝐴 ∖ 𝑊) ↔ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑊))
2		suppssrg.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
3	2	ffnd 6664	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
4		suppssrg.a	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
5		suppssrg.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
6		elsuppfng 8113	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑈) → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍)))
7	3, 4, 5, 6	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍)))
8		suppssrg.n	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)
9	8	sseld 3921	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) → 𝑋 ∈ 𝑊))
10	7, 9	sylbird 260	. . . . 5 ⊢ (𝜑 → ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍) → 𝑋 ∈ 𝑊))
11	10	expdimp 452	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ≠ 𝑍 → 𝑋 ∈ 𝑊))
12	11	necon1bd 2951	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (¬ 𝑋 ∈ 𝑊 → (𝐹‘𝑋) = 𝑍))
13	12	impr 454	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑊)) → (𝐹‘𝑋) = 𝑍)
14	1, 13	sylan2b 595	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   ∖ cdif 3887   ⊆ wss 3890   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361   supp csupp 8104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-supp 8105

This theorem is referenced by:  psrbaglesupp  21915  psrbaglefi  21919  evlsvvvallem2  22083  mhpmulcl  22128  mhpvscacl  22133  selvvvval  43035