Theorem fsuppcurry2 30963

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 7262	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥𝐹𝐶) = (𝑦𝐹𝐶))
3	2	cbvmptv 5183	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶)) = (𝑦 ∈ 𝐴 ↦ (𝑦𝐹𝐶))
4	1, 3	eqtri 2766	. . 3 ⊢ 𝐺 = (𝑦 ∈ 𝐴 ↦ (𝑦𝐹𝐶))
5		fsuppcurry2.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6	5	mptexd 7082	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ (𝑦𝐹𝐶)) ∈ V)
7	4, 6	eqeltrid 2843	. 2 ⊢ (𝜑 → 𝐺 ∈ V)
8	1	funmpt2 6457	. . 3 ⊢ Fun 𝐺
9	8	a1i 11	. 2 ⊢ (𝜑 → Fun 𝐺)
10		fsuppcurry2.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
11		fo1st 7824	. . . . 5 ⊢ 1st :V–onto→V
12		fofun 6673	. . . . 5 ⊢ (1st :V–onto→V → Fun 1st )
13	11, 12	ax-mp 5	. . . 4 ⊢ Fun 1st
14		funres 6460	. . . 4 ⊢ (Fun 1st → Fun (1st ↾ (V × V)))
15	13, 14	mp1i 13	. . 3 ⊢ (𝜑 → Fun (1st ↾ (V × V)))
16		fsuppcurry2.1	. . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍)
17	16	fsuppimpd 9065	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
18		imafi 8920	. . 3 ⊢ ((Fun (1st ↾ (V × V)) ∧ (𝐹 supp 𝑍) ∈ Fin) → ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) ∈ Fin)
19	15, 17, 18	syl2anc 583	. 2 ⊢ (𝜑 → ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) ∈ Fin)
20		ovexd 7290	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑦𝐹𝐶) ∈ V)
21	20, 4	fmptd 6970	. . 3 ⊢ (𝜑 → 𝐺:𝐴⟶V)
22		eldif 3893	. . . 4 ⊢ (𝑦 ∈ (𝐴 ∖ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍))) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍))))
23		simplr 765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → 𝑦 ∈ 𝐴)
24		fsuppcurry2.c	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
25	24	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → 𝐶 ∈ 𝐵)
26	23, 25	opelxpd 5618	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵))
27		df-ov 7258	. . . . . . . . . . 11 ⊢ (𝑦𝐹𝐶) = (𝐹‘⟨𝑦, 𝐶⟩)
28		ovexd 7290	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝑦𝐹𝐶) ∈ V)
29	1, 2, 23, 28	fvmptd3 6880	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝐺‘𝑦) = (𝑦𝐹𝐶))
30		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ¬ (𝐺‘𝑦) = 𝑍)
31	30	neqned 2949	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝐺‘𝑦) ≠ 𝑍)
32	29, 31	eqnetrrd 3011	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝑦𝐹𝐶) ≠ 𝑍)
33	27, 32	eqnetrrid 3018	. . . . . . . . . 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 7579	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
37		elsuppfn 7958	. . . . . . . . . . . 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 1369	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨𝑦, 𝐶⟩ ∈ (𝐹 supp 𝑍) ↔ (⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵) ∧ (𝐹‘⟨𝑦, 𝐶⟩) ≠ 𝑍)))
39	38	ad2antrr 722	. . . . . . . . . 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 709	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ⟨𝑦, 𝐶⟩ ∈ (𝐹 supp 𝑍))
41		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → 𝑧 = ⟨𝑦, 𝐶⟩)
42	41	fveq2d 6760	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ((1st ↾ (V × V))‘𝑧) = ((1st ↾ (V × V))‘⟨𝑦, 𝐶⟩))
43		xpss 5596	. . . . . . . . . . . 12 ⊢ (𝐴 × 𝐵) ⊆ (V × V)
44	26	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵))
45	43, 44	sselid 3915	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ⟨𝑦, 𝐶⟩ ∈ (V × V))
46	45	fvresd 6776	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ((1st ↾ (V × V))‘⟨𝑦, 𝐶⟩) = (1st ‘⟨𝑦, 𝐶⟩))
47	23	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → 𝑦 ∈ 𝐴)
48	25	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → 𝐶 ∈ 𝐵)
49		op1stg 7816	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (1st ‘⟨𝑦, 𝐶⟩) = 𝑦)
50	47, 48, 49	syl2anc 583	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → (1st ‘⟨𝑦, 𝐶⟩) = 𝑦)
51	42, 46, 50	3eqtrd 2782	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ((1st ↾ (V × V))‘𝑧) = 𝑦)
52	40, 51	rspcedeq1vd 3558	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ∃𝑧 ∈ (𝐹 supp 𝑍)((1st ↾ (V × V))‘𝑧) = 𝑦)
53		fofn 6674	. . . . . . . . . . . . 13 ⊢ (1st :V–onto→V → 1st Fn V)
54		fnresin 30862	. . . . . . . . . . . . 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 3941	. . . . . . . . . . . . . 14 ⊢ (V × V) ⊆ V
57		sseqin2 4146	. . . . . . . . . . . . . 14 ⊢ ((V × V) ⊆ V ↔ (V ∩ (V × V)) = (V × V))
58	56, 57	mpbi 229	. . . . . . . . . . . . 13 ⊢ (V ∩ (V × V)) = (V × V)
59	58	fneq2i 6515	. . . . . . . . . . . 12 ⊢ ((1st ↾ (V × V)) Fn (V ∩ (V × V)) ↔ (1st ↾ (V × V)) Fn (V × V))
60	55, 59	mpbi 229	. . . . . . . . . . 11 ⊢ (1st ↾ (V × V)) Fn (V × V)
61	60	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (1st ↾ (V × V)) Fn (V × V))
62		suppssdm 7964	. . . . . . . . . . . 12 ⊢ (𝐹 supp 𝑍) ⊆ dom 𝐹
63	34	fndmd 6522	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐹 = (𝐴 × 𝐵))
64	62, 63	sseqtrid 3969	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐴 × 𝐵))
65	64, 43	sstrdi 3929	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (V × V))
66	61, 65	fvelimabd 6824	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) ↔ ∃𝑧 ∈ (𝐹 supp 𝑍)((1st ↾ (V × V))‘𝑧) = 𝑦))
67	66	ad2antrr 722	. . . . . . . 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 256	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))
69	68	ex 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (¬ (𝐺‘𝑦) = 𝑍 → 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍))))
70	69	con1d 145	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (¬ 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) → (𝐺‘𝑦) = 𝑍))
71	70	impr 454	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))) → (𝐺‘𝑦) = 𝑍)
72	22, 71	sylan2b 593	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))) → (𝐺‘𝑦) = 𝑍)
73	21, 72	suppss 7981	. 2 ⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))
74		suppssfifsupp 9073	. 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 1376	1 ⊢ (𝜑 → 𝐺 finSupp 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064  Vcvv 3422   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ⟨cop 4564   class class class wbr 5070   ↦ cmpt 5153   × cxp 5578  dom cdm 5580   ↾ cres 5582   “ cima 5583  Fun wfun 6412   Fn wfn 6413  –onto→wfo 6416  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802   supp csupp 7948  Fincfn 8691   finSupp cfsupp 9058

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-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  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-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  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-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-supp 7949  df-1o 8267  df-en 8692  df-fin 8695  df-fsupp 9059

This theorem is referenced by: (None)