Theorem suppssrg 8138

Description: A function is zero outside its support. Version of suppssr 8137 avoiding ax-rep 5224 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 3911	. 2 ⊢ (𝑋 ∈ (𝐴 ∖ 𝑊) ↔ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑊))
2		suppssrg.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
3	2	ffnd 6663	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
4		suppssrg.a	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
5		suppssrg.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
6		elsuppfng 8111	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑈) → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍)))
7	3, 4, 5, 6	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍)))
8		suppssrg.n	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)
9	8	sseld 3932	. . . . . 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 2950	. . 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 1541   ∈ wcel 2113   ≠ wne 2932   ∖ cdif 3898   ⊆ wss 3901   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   supp csupp 8102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-supp 8103

This theorem is referenced by:  psrbaglesupp  21878  psrbaglefi  21882  evlsvvvallem2  22047  mhpmulcl  22092  mhpvscacl  22097  selvvvval  42838