Theorem suppssrg 7984

Description: A function is zero outside its support. Version of suppssr 7983 avoiding ax-rep 5205 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 3893	. 2 ⊢ (𝑋 ∈ (𝐴 ∖ 𝑊) ↔ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑊))
2		suppssrg.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
3	2	ffnd 6585	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
4		suppssrg.a	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
5		suppssrg.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
6		elsuppfng 7957	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑈) → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍)))
7	3, 4, 5, 6	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍)))
8		suppssrg.n	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)
9	8	sseld 3916	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) → 𝑋 ∈ 𝑊))
10	7, 9	sylbird 259	. . . . 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 2960	. . 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 593	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942   ∖ cdif 3880   ⊆ wss 3883   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   supp csupp 7948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-supp 7949

This theorem is referenced by:  psrbaglesupp  21037  psrbaglefi  21045