Theorem fsuppcurry1 32214

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

Hypotheses

Ref	Expression
fsuppcurry1.g	⊢ 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))
fsuppcurry1.z	⊢ (𝜑 → 𝑍 ∈ 𝑈)
fsuppcurry1.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fsuppcurry1.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
fsuppcurry1.f	⊢ (𝜑 → 𝐹 Fn (𝐴 × 𝐵))
fsuppcurry1.c	⊢ (𝜑 → 𝐶 ∈ 𝐴)
fsuppcurry1.1	⊢ (𝜑 → 𝐹 finSupp 𝑍)

Assertion

Ref	Expression
fsuppcurry1	⊢ (𝜑 → 𝐺 finSupp 𝑍)

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

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

Proof of Theorem fsuppcurry1

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

Step	Hyp	Ref	Expression
1		fsuppcurry1.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))
2		oveq2 7420	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐶𝐹𝑥) = (𝐶𝐹𝑦))
3	2	cbvmptv 5262	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)) = (𝑦 ∈ 𝐵 ↦ (𝐶𝐹𝑦))
4	1, 3	eqtri 2759	. . 3 ⊢ 𝐺 = (𝑦 ∈ 𝐵 ↦ (𝐶𝐹𝑦))
5		fsuppcurry1.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
6	5	mptexd 7229	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ (𝐶𝐹𝑦)) ∈ V)
7	4, 6	eqeltrid 2836	. 2 ⊢ (𝜑 → 𝐺 ∈ V)
8	1	funmpt2 6588	. . 3 ⊢ Fun 𝐺
9	8	a1i 11	. 2 ⊢ (𝜑 → Fun 𝐺)
10		fsuppcurry1.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
11		fo2nd 7999	. . . . 5 ⊢ 2nd :V–onto→V
12		fofun 6807	. . . . 5 ⊢ (2nd :V–onto→V → Fun 2nd )
13	11, 12	ax-mp 5	. . . 4 ⊢ Fun 2nd
14		funres 6591	. . . 4 ⊢ (Fun 2nd → Fun (2nd ↾ (V × V)))
15	13, 14	mp1i 13	. . 3 ⊢ (𝜑 → Fun (2nd ↾ (V × V)))
16		fsuppcurry1.1	. . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍)
17	16	fsuppimpd 9372	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
18		imafi 9178	. . 3 ⊢ ((Fun (2nd ↾ (V × V)) ∧ (𝐹 supp 𝑍) ∈ Fin) → ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)) ∈ Fin)
19	15, 17, 18	syl2anc 583	. 2 ⊢ (𝜑 → ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)) ∈ Fin)
20		ovexd 7447	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐶𝐹𝑦) ∈ V)
21	20, 4	fmptd 7116	. . 3 ⊢ (𝜑 → 𝐺:𝐵⟶V)
22		eldif 3959	. . . 4 ⊢ (𝑦 ∈ (𝐵 ∖ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍))) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍))))
23		fsuppcurry1.c	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
24	23	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → 𝐶 ∈ 𝐴)
25		simplr 766	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → 𝑦 ∈ 𝐵)
26	24, 25	opelxpd 5716	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ⟨𝐶, 𝑦⟩ ∈ (𝐴 × 𝐵))
27		df-ov 7415	. . . . . . . . . . 11 ⊢ (𝐶𝐹𝑦) = (𝐹‘⟨𝐶, 𝑦⟩)
28		ovexd 7447	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝐶𝐹𝑦) ∈ V)
29	1, 2, 25, 28	fvmptd3 7022	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝐺‘𝑦) = (𝐶𝐹𝑦))
30		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ¬ (𝐺‘𝑦) = 𝑍)
31	30	neqned 2946	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝐺‘𝑦) ≠ 𝑍)
32	29, 31	eqnetrrd 3008	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝐶𝐹𝑦) ≠ 𝑍)
33	27, 32	eqnetrrid 3015	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝐹‘⟨𝐶, 𝑦⟩) ≠ 𝑍)
34		fsuppcurry1.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 Fn (𝐴 × 𝐵))
35		fsuppcurry1.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
36	35, 5	xpexd 7741	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
37		elsuppfn 8159	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ∈ V ∧ 𝑍 ∈ 𝑈) → (⟨𝐶, 𝑦⟩ ∈ (𝐹 supp 𝑍) ↔ (⟨𝐶, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ (𝐹‘⟨𝐶, 𝑦⟩) ≠ 𝑍)))
38	34, 36, 10, 37	syl3anc 1370	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨𝐶, 𝑦⟩ ∈ (𝐹 supp 𝑍) ↔ (⟨𝐶, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ (𝐹‘⟨𝐶, 𝑦⟩) ≠ 𝑍)))
39	38	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (⟨𝐶, 𝑦⟩ ∈ (𝐹 supp 𝑍) ↔ (⟨𝐶, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ (𝐹‘⟨𝐶, 𝑦⟩) ≠ 𝑍)))
40	26, 33, 39	mpbir2and 710	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ⟨𝐶, 𝑦⟩ ∈ (𝐹 supp 𝑍))
41		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → 𝑧 = ⟨𝐶, 𝑦⟩)
42	41	fveq2d 6896	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → ((2nd ↾ (V × V))‘𝑧) = ((2nd ↾ (V × V))‘⟨𝐶, 𝑦⟩))
43		xpss 5693	. . . . . . . . . . . 12 ⊢ (𝐴 × 𝐵) ⊆ (V × V)
44	26	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → ⟨𝐶, 𝑦⟩ ∈ (𝐴 × 𝐵))
45	43, 44	sselid 3981	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → ⟨𝐶, 𝑦⟩ ∈ (V × V))
46	45	fvresd 6912	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → ((2nd ↾ (V × V))‘⟨𝐶, 𝑦⟩) = (2nd ‘⟨𝐶, 𝑦⟩))
47	24	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → 𝐶 ∈ 𝐴)
48	25	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → 𝑦 ∈ 𝐵)
49		op2ndg 7991	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (2nd ‘⟨𝐶, 𝑦⟩) = 𝑦)
50	47, 48, 49	syl2anc 583	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → (2nd ‘⟨𝐶, 𝑦⟩) = 𝑦)
51	42, 46, 50	3eqtrd 2775	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → ((2nd ↾ (V × V))‘𝑧) = 𝑦)
52	40, 51	rspcedeq1vd 3619	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ∃𝑧 ∈ (𝐹 supp 𝑍)((2nd ↾ (V × V))‘𝑧) = 𝑦)
53		fofn 6808	. . . . . . . . . . . . 13 ⊢ (2nd :V–onto→V → 2nd Fn V)
54		fnresin 32114	. . . . . . . . . . . . 13 ⊢ (2nd Fn V → (2nd ↾ (V × V)) Fn (V ∩ (V × V)))
55	11, 53, 54	mp2b 10	. . . . . . . . . . . 12 ⊢ (2nd ↾ (V × V)) Fn (V ∩ (V × V))
56		ssv 4007	. . . . . . . . . . . . . 14 ⊢ (V × V) ⊆ V
57		sseqin2 4216	. . . . . . . . . . . . . 14 ⊢ ((V × V) ⊆ V ↔ (V ∩ (V × V)) = (V × V))
58	56, 57	mpbi 229	. . . . . . . . . . . . 13 ⊢ (V ∩ (V × V)) = (V × V)
59	58	fneq2i 6648	. . . . . . . . . . . 12 ⊢ ((2nd ↾ (V × V)) Fn (V ∩ (V × V)) ↔ (2nd ↾ (V × V)) Fn (V × V))
60	55, 59	mpbi 229	. . . . . . . . . . 11 ⊢ (2nd ↾ (V × V)) Fn (V × V)
61	60	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (2nd ↾ (V × V)) Fn (V × V))
62		suppssdm 8165	. . . . . . . . . . . 12 ⊢ (𝐹 supp 𝑍) ⊆ dom 𝐹
63	34	fndmd 6655	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐹 = (𝐴 × 𝐵))
64	62, 63	sseqtrid 4035	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐴 × 𝐵))
65	64, 43	sstrdi 3995	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (V × V))
66	61, 65	fvelimabd 6966	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)) ↔ ∃𝑧 ∈ (𝐹 supp 𝑍)((2nd ↾ (V × V))‘𝑧) = 𝑦))
67	66	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝑦 ∈ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)) ↔ ∃𝑧 ∈ (𝐹 supp 𝑍)((2nd ↾ (V × V))‘𝑧) = 𝑦))
68	52, 67	mpbird 256	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → 𝑦 ∈ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)))
69	68	ex 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (¬ (𝐺‘𝑦) = 𝑍 → 𝑦 ∈ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍))))
70	69	con1d 145	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (¬ 𝑦 ∈ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)) → (𝐺‘𝑦) = 𝑍))
71	70	impr 454	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)))) → (𝐺‘𝑦) = 𝑍)
72	22, 71	sylan2b 593	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∖ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)))) → (𝐺‘𝑦) = 𝑍)
73	21, 72	suppss 8182	. 2 ⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)))
74		suppssfifsupp 9381	. 2 ⊢ (((𝐺 ∈ V ∧ Fun 𝐺 ∧ 𝑍 ∈ 𝑈) ∧ (((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)) ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)))) → 𝐺 finSupp 𝑍)
75	7, 9, 10, 19, 73, 74	syl32anc 1377	1 ⊢ (𝜑 → 𝐺 finSupp 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  ∃wrex 3069  Vcvv 3473   ∖ cdif 3946   ∩ cin 3948   ⊆ wss 3949  ⟨cop 4635   class class class wbr 5149   ↦ cmpt 5232   × cxp 5675  dom cdm 5677   ↾ cres 5679   “ cima 5680  Fun wfun 6538   Fn wfn 6539  –onto→wfo 6542  ‘cfv 6544  (class class class)co 7412  2^nd c2nd 7977   supp csupp 8149  Fincfn 8942   finSupp cfsupp 9364

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-2nd 7979  df-supp 8150  df-1o 8469  df-en 8943  df-fin 8946  df-fsupp 9365

This theorem is referenced by:  fedgmullem2  33000