Theorem fsuppssov1 9299

Description: Formula building theorem for finite support: operator with left annihilator. Finite support version of suppssov1 8149. (Contributed by SN, 26-Apr-2025.)

Hypotheses

Ref	Expression
fsuppssov1.s	⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ 𝐴) finSupp 𝑌)
fsuppssov1.o	⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)
fsuppssov1.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑉)
fsuppssov1.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑅)
fsuppssov1.z	⊢ (𝜑 → 𝑍 ∈ 𝑊)

Assertion

Ref	Expression
fsuppssov1	⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) finSupp 𝑍)

Distinct variable groups:   𝜑,𝑣   𝜑,𝑥   𝑣,𝐵   𝑥,𝐷   𝑣,𝑂   𝑣,𝑅   𝑣,𝑌   𝑥,𝑌   𝑣,𝑍   𝑥,𝑍   𝑥,𝑉

Allowed substitution hints:   𝐴(𝑥,𝑣)   𝐵(𝑥)   𝐷(𝑣)   𝑅(𝑥)   𝑂(𝑥)   𝑉(𝑣)   𝑊(𝑥,𝑣)

Proof of Theorem fsuppssov1

Step	Hyp	Ref	Expression
1		fsuppssov1.s	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ 𝐴) finSupp 𝑌)
2		relfsupp 9278	. . . . . 6 ⊢ Rel finSupp
3	2	brrelex1i 5688	. . . . 5 ⊢ ((𝑥 ∈ 𝐷 ↦ 𝐴) finSupp 𝑌 → (𝑥 ∈ 𝐷 ↦ 𝐴) ∈ V)
4	1, 3	syl 17	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ 𝐴) ∈ V)
5		fsuppssov1.a	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑉)
6	5	fmpttd 7069	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ 𝐴):𝐷⟶𝑉)
7		dmfex 7857	. . . 4 ⊢ (((𝑥 ∈ 𝐷 ↦ 𝐴) ∈ V ∧ (𝑥 ∈ 𝐷 ↦ 𝐴):𝐷⟶𝑉) → 𝐷 ∈ V)
8	4, 6, 7	syl2anc 585	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
9	8	mptexd 7180	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V)
10		fsuppssov1.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
11		funmpt 6538	. . 3 ⊢ Fun (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵))
12	11	a1i 11	. 2 ⊢ (𝜑 → Fun (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)))
13		ssidd 3959	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌) ⊆ ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌))
14		fsuppssov1.o	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)
15		fsuppssov1.b	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑅)
16	2	brrelex2i 5689	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ↦ 𝐴) finSupp 𝑌 → 𝑌 ∈ V)
17	1, 16	syl 17	. . 3 ⊢ (𝜑 → 𝑌 ∈ V)
18	13, 14, 5, 15, 17	suppssov1 8149	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌))
19	9, 10, 12, 1, 18	fsuppsssuppgd 9297	1 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) finSupp 𝑍)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442   class class class wbr 5100   ↦ cmpt 5181  Fun wfun 6494  ⟶wf 6496  (class class class)co 7368   supp csupp 8112   finSupp cfsupp 9276

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-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-supp 8113  df-1o 8407  df-en 8896  df-fin 8899  df-fsupp 9277

This theorem is referenced by:  elrgspn  33339  elrgspnsubrunlem2  33341  evlextv  33718  mplvrpmrhm  33723  selvvvval  42937  evlselv  42939