Theorem suppssrg 8219

Description: A function is zero outside its support. Version of suppssr 8218 avoiding ax-rep 5284 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 3972	. 2 ⊢ (𝑋 ∈ (𝐴 ∖ 𝑊) ↔ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑊))
2		suppssrg.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
3	2	ffnd 6737	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
4		suppssrg.a	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
5		suppssrg.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
6		elsuppfng 8192	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑈) → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍)))
7	3, 4, 5, 6	syl3anc 1370	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍)))
8		suppssrg.n	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)
9	8	sseld 3993	. . . . . 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 2955	. . 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 594	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1536   ∈ wcel 2105   ≠ wne 2937   ∖ cdif 3959   ⊆ wss 3962   Fn wfn 6557  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430   supp csupp 8183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-supp 8184

This theorem is referenced by:  psrbaglesupp  21959  psrbaglefi  21963  mhpmulcl  22170  mhpvscacl  22175  evlsvvvallem2  42548  selvvvval  42571