Theorem fsuppcurry2 32982

Description: Finite support of a curried function with a constant second argument. (Contributed by Thierry Arnoux, 7-Jul-2023.)

Hypotheses

Ref	Expression
fsuppcurry2.g	⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))
fsuppcurry2.z	⊢ (𝜑 → 𝑍 ∈ 𝑈)
fsuppcurry2.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fsuppcurry2.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
fsuppcurry2.f	⊢ (𝜑 → 𝐹 Fn (𝐴 × 𝐵))
fsuppcurry2.c	⊢ (𝜑 → 𝐶 ∈ 𝐵)
fsuppcurry2.1	⊢ (𝜑 → 𝐹 finSupp 𝑍)

Assertion

Ref	Expression
fsuppcurry2	⊢ (𝜑 → 𝐺 finSupp 𝑍)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐹

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝑈(𝑥)   𝐺(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑍(𝑥)

Proof of Theorem fsuppcurry2

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsuppcurry2.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))
2		oveq1 7407	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥𝐹𝐶) = (𝑦𝐹𝐶))
3	2	cbvmptv 5209	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶)) = (𝑦 ∈ 𝐴 ↦ (𝑦𝐹𝐶))
4	1, 3	eqtri 2788	. . 3 ⊢ 𝐺 = (𝑦 ∈ 𝐴 ↦ (𝑦𝐹𝐶))
5		fsuppcurry2.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6	5	mptexd 7212	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ (𝑦𝐹𝐶)) ∈ V)
7	4, 6	eqeltrid 2869	. 2 ⊢ (𝜑 → 𝐺 ∈ V)
8	1	funmpt2 6564	. . 3 ⊢ Fun 𝐺
9	8	a1i 11	. 2 ⊢ (𝜑 → Fun 𝐺)
10		fsuppcurry2.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
11		fo1st 7994	. . . . 5 ⊢ 1st :V–onto→V
12		fofun 6783	. . . . 5 ⊢ (1st :V–onto→V → Fun 1st )
13	11, 12	ax-mp 5	. . . 4 ⊢ Fun 1st
14		funres 6567	. . . 4 ⊢ (Fun 1st → Fun (1st ↾ (V × V)))
15	13, 14	mp1i 14	. . 3 ⊢ (𝜑 → Fun (1st ↾ (V × V)))
16		fsuppcurry2.1	. . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍)
17	16	fsuppimpd 9317	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
18		imafi 9263	. . 3 ⊢ ((Fun (1st ↾ (V × V)) ∧ (𝐹 supp 𝑍) ∈ Fin) → ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) ∈ Fin)
19	15, 17, 18	syl2anc 595	. 2 ⊢ (𝜑 → ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) ∈ Fin)
20		ovexd 7435	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑦𝐹𝐶) ∈ V)
21	20, 4	fmptd 7099	. . 3 ⊢ (𝜑 → 𝐺:𝐴⟶V)
22		eldif 3917	. . . 4 ⊢ (𝑦 ∈ (𝐴 ∖ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍))) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍))))
23		simplr 780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → 𝑦 ∈ 𝐴)
24		fsuppcurry2.c	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
25	24	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → 𝐶 ∈ 𝐵)
26	23, 25	opelxpd 5691	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵))
27		df-ov 7403	. . . . . . . . . . 11 ⊢ (𝑦𝐹𝐶) = (𝐹‘⟨𝑦, 𝐶⟩)
28		ovexd 7435	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝑦𝐹𝐶) ∈ V)
29	1, 2, 23, 28	fvmptd3 7003	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝐺‘𝑦) = (𝑦𝐹𝐶))
30		simpr 489	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ¬ (𝐺‘𝑦) = 𝑍)
31	30	neqned 2967	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝐺‘𝑦) ≠ 𝑍)
32	29, 31	eqnetrrd 3028	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝑦𝐹𝐶) ≠ 𝑍)
33	27, 32	eqnetrrid 3035	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝐹‘⟨𝑦, 𝐶⟩) ≠ 𝑍)
34		fsuppcurry2.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 Fn (𝐴 × 𝐵))
35		fsuppcurry2.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
36	5, 35	xpexd 7738	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
37		elsuppfn 8154	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ∈ V ∧ 𝑍 ∈ 𝑈) → (⟨𝑦, 𝐶⟩ ∈ (𝐹 supp 𝑍) ↔ (⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵) ∧ (𝐹‘⟨𝑦, 𝐶⟩) ≠ 𝑍)))
38	34, 36, 10, 37	syl3anc 1394	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨𝑦, 𝐶⟩ ∈ (𝐹 supp 𝑍) ↔ (⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵) ∧ (𝐹‘⟨𝑦, 𝐶⟩) ≠ 𝑍)))
39	38	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (⟨𝑦, 𝐶⟩ ∈ (𝐹 supp 𝑍) ↔ (⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵) ∧ (𝐹‘⟨𝑦, 𝐶⟩) ≠ 𝑍)))
40	26, 33, 39	mpbir2and 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ⟨𝑦, 𝐶⟩ ∈ (𝐹 supp 𝑍))
41		simpr 489	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → 𝑧 = ⟨𝑦, 𝐶⟩)
42	41	fveq2d 6875	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ((1st ↾ (V × V))‘𝑧) = ((1st ↾ (V × V))‘⟨𝑦, 𝐶⟩))
43		xpss 5668	. . . . . . . . . . . 12 ⊢ (𝐴 × 𝐵) ⊆ (V × V)
44	26	adantr 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵))
45	43, 44	sselid 3937	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ⟨𝑦, 𝐶⟩ ∈ (V × V))
46	45	fvresd 6891	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ((1st ↾ (V × V))‘⟨𝑦, 𝐶⟩) = (1st ‘⟨𝑦, 𝐶⟩))
47	23	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → 𝑦 ∈ 𝐴)
48	25	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → 𝐶 ∈ 𝐵)
49		op1stg 7986	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (1st ‘⟨𝑦, 𝐶⟩) = 𝑦)
50	47, 48, 49	syl2anc 595	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → (1st ‘⟨𝑦, 𝐶⟩) = 𝑦)
51	42, 46, 50	3eqtrd 2804	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ((1st ↾ (V × V))‘𝑧) = 𝑦)
52	40, 51	rspcedeq1vd 3591	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ∃𝑧 ∈ (𝐹 supp 𝑍)((1st ↾ (V × V))‘𝑧) = 𝑦)
53		fofn 6784	. . . . . . . . . . . . 13 ⊢ (1st :V–onto→V → 1st Fn V)
54		fnresin 32881	. . . . . . . . . . . . 13 ⊢ (1st Fn V → (1st ↾ (V × V)) Fn (V ∩ (V × V)))
55	11, 53, 54	mp2b 10	. . . . . . . . . . . 12 ⊢ (1st ↾ (V × V)) Fn (V ∩ (V × V))
56		ssv 3963	. . . . . . . . . . . . . 14 ⊢ (V × V) ⊆ V
57		sseqin2 4178	. . . . . . . . . . . . . 14 ⊢ ((V × V) ⊆ V ↔ (V ∩ (V × V)) = (V × V))
58	56, 57	mpbi 233	. . . . . . . . . . . . 13 ⊢ (V ∩ (V × V)) = (V × V)
59	58	fneq2i 6623	. . . . . . . . . . . 12 ⊢ ((1st ↾ (V × V)) Fn (V ∩ (V × V)) ↔ (1st ↾ (V × V)) Fn (V × V))
60	55, 59	mpbi 233	. . . . . . . . . . 11 ⊢ (1st ↾ (V × V)) Fn (V × V)
61	60	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (1st ↾ (V × V)) Fn (V × V))
62		suppssdm 8161	. . . . . . . . . . . 12 ⊢ (𝐹 supp 𝑍) ⊆ dom 𝐹
63	34	fndmd 6630	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐹 = (𝐴 × 𝐵))
64	62, 63	sseqtrid 3981	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐴 × 𝐵))
65	64, 43	sstrdi 3951	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (V × V))
66	61, 65	fvelimabd 6944	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) ↔ ∃𝑧 ∈ (𝐹 supp 𝑍)((1st ↾ (V × V))‘𝑧) = 𝑦))
67	66	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) ↔ ∃𝑧 ∈ (𝐹 supp 𝑍)((1st ↾ (V × V))‘𝑧) = 𝑦))
68	52, 67	mpbird 260	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))
69	68	ex 417	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (¬ (𝐺‘𝑦) = 𝑍 → 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍))))
70	69	con1d 146	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (¬ 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) → (𝐺‘𝑦) = 𝑍))
71	70	impr 459	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))) → (𝐺‘𝑦) = 𝑍)
72	22, 71	sylan2b 605	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))) → (𝐺‘𝑦) = 𝑍)
73	21, 72	suppss 8178	. 2 ⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))
74		suppssfifsupp 9328	. 2 ⊢ (((𝐺 ∈ V ∧ Fun 𝐺 ∧ 𝑍 ∈ 𝑈) ∧ (((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))) → 𝐺 finSupp 𝑍)
75	7, 9, 10, 19, 73, 74	syl32anc 1401	1 ⊢ (𝜑 → 𝐺 finSupp 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∃wrex 3089  Vcvv 3457   ∖ cdif 3904   ∩ cin 3906   ⊆ wss 3907  ⟨cop 4591   class class class wbr 5105   ↦ cmpt 5186   × cxp 5650  dom cdm 5652   ↾ cres 5654   “ cima 5655  Fun wfun 6519   Fn wfn 6520  –onto→wfo 6523  ‘cfv 6525  (class class class)co 7400  1^st c1st 7972   supp csupp 8144  Fincfn 8931   finSupp cfsupp 9309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-supp 8145  df-1o 8441  df-en 8932  df-dom 8933  df-fin 8935  df-fsupp 9310

This theorem is referenced by: (None)