Theorem fsuppcurry1 33069

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 7418	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐶𝐹𝑥) = (𝐶𝐹𝑦))
3	2	cbvmptv 5215	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)) = (𝑦 ∈ 𝐵 ↦ (𝐶𝐹𝑦))
4	1, 3	eqtri 2786	. . 3 ⊢ 𝐺 = (𝑦 ∈ 𝐵 ↦ (𝐶𝐹𝑦))
5		fsuppcurry1.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
6	5	mptexd 7222	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ (𝐶𝐹𝑦)) ∈ V)
7	4, 6	eqeltrid 2867	. 2 ⊢ (𝜑 → 𝐺 ∈ V)
8	1	funmpt2 6575	. . 3 ⊢ Fun 𝐺
9	8	a1i 11	. 2 ⊢ (𝜑 → Fun 𝐺)
10		fsuppcurry1.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
11		fo2nd 8003	. . . . 5 ⊢ 2nd :V–onto→V
12		fofun 6793	. . . . 5 ⊢ (2nd :V–onto→V → Fun 2nd )
13	11, 12	ax-mp 5	. . . 4 ⊢ Fun 2nd
14		funres 6578	. . . 4 ⊢ (Fun 2nd → Fun (2nd ↾ (V × V)))
15	13, 14	mp1i 14	. . 3 ⊢ (𝜑 → Fun (2nd ↾ (V × V)))
16		fsuppcurry1.1	. . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍)
17	16	fsuppimpd 9325	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
18		imafi 9271	. . 3 ⊢ ((Fun (2nd ↾ (V × V)) ∧ (𝐹 supp 𝑍) ∈ Fin) → ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)) ∈ Fin)
19	15, 17, 18	syl2anc 595	. 2 ⊢ (𝜑 → ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)) ∈ Fin)
20		ovexd 7445	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐶𝐹𝑦) ∈ V)
21	20, 4	fmptd 7109	. . 3 ⊢ (𝜑 → 𝐺:𝐵⟶V)
22		eldif 3915	. . . 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 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → 𝐶 ∈ 𝐴)
25		simplr 780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → 𝑦 ∈ 𝐵)
26	24, 25	opelxpd 5700	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ⟨𝐶, 𝑦⟩ ∈ (𝐴 × 𝐵))
27		df-ov 7413	. . . . . . . . . . 11 ⊢ (𝐶𝐹𝑦) = (𝐹‘⟨𝐶, 𝑦⟩)
28		ovexd 7445	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝐶𝐹𝑦) ∈ V)
29	1, 2, 25, 28	fvmptd3 7013	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝐺‘𝑦) = (𝐶𝐹𝑦))
30		simpr 489	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ¬ (𝐺‘𝑦) = 𝑍)
31	30	neqned 2965	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝐺‘𝑦) ≠ 𝑍)
32	29, 31	eqnetrrd 3026	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝐶𝐹𝑦) ≠ 𝑍)
33	27, 32	eqnetrrid 3033	. . . . . . . . . 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 7746	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
37		elsuppfn 8162	. . . . . . . . . . . 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 1398	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨𝐶, 𝑦⟩ ∈ (𝐹 supp 𝑍) ↔ (⟨𝐶, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ (𝐹‘⟨𝐶, 𝑦⟩) ≠ 𝑍)))
39	38	ad2antrr 738	. . . . . . . . . 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 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ⟨𝐶, 𝑦⟩ ∈ (𝐹 supp 𝑍))
41		simpr 489	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → 𝑧 = ⟨𝐶, 𝑦⟩)
42	41	fveq2d 6885	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → ((2nd ↾ (V × V))‘𝑧) = ((2nd ↾ (V × V))‘⟨𝐶, 𝑦⟩))
43		xpss 5677	. . . . . . . . . . . 12 ⊢ (𝐴 × 𝐵) ⊆ (V × V)
44	26	adantr 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → ⟨𝐶, 𝑦⟩ ∈ (𝐴 × 𝐵))
45	43, 44	sselid 3935	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → ⟨𝐶, 𝑦⟩ ∈ (V × V))
46	45	fvresd 6901	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → ((2nd ↾ (V × V))‘⟨𝐶, 𝑦⟩) = (2nd ‘⟨𝐶, 𝑦⟩))
47	24	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → 𝐶 ∈ 𝐴)
48	25	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → 𝑦 ∈ 𝐵)
49		op2ndg 7995	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (2nd ‘⟨𝐶, 𝑦⟩) = 𝑦)
50	47, 48, 49	syl2anc 595	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → (2nd ‘⟨𝐶, 𝑦⟩) = 𝑦)
51	42, 46, 50	3eqtrd 2802	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → ((2nd ↾ (V × V))‘𝑧) = 𝑦)
52	40, 51	rspcedeq1vd 3588	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ∃𝑧 ∈ (𝐹 supp 𝑍)((2nd ↾ (V × V))‘𝑧) = 𝑦)
53		fofn 6794	. . . . . . . . . . . . 13 ⊢ (2nd :V–onto→V → 2nd Fn V)
54		fnresin 32969	. . . . . . . . . . . . 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 3961	. . . . . . . . . . . . . 14 ⊢ (V × V) ⊆ V
57		sseqin2 4176	. . . . . . . . . . . . . 14 ⊢ ((V × V) ⊆ V ↔ (V ∩ (V × V)) = (V × V))
58	56, 57	mpbi 233	. . . . . . . . . . . . 13 ⊢ (V ∩ (V × V)) = (V × V)
59	58	fneq2i 6633	. . . . . . . . . . . 12 ⊢ ((2nd ↾ (V × V)) Fn (V ∩ (V × V)) ↔ (2nd ↾ (V × V)) Fn (V × V))
60	55, 59	mpbi 233	. . . . . . . . . . 11 ⊢ (2nd ↾ (V × V)) Fn (V × V)
61	60	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (2nd ↾ (V × V)) Fn (V × V))
62		suppssdm 8169	. . . . . . . . . . . 12 ⊢ (𝐹 supp 𝑍) ⊆ dom 𝐹
63	34	fndmd 6640	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐹 = (𝐴 × 𝐵))
64	62, 63	sseqtrid 3979	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐴 × 𝐵))
65	64, 43	sstrdi 3949	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (V × V))
66	61, 65	fvelimabd 6954	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)) ↔ ∃𝑧 ∈ (𝐹 supp 𝑍)((2nd ↾ (V × V))‘𝑧) = 𝑦))
67	66	ad2antrr 738	. . . . . . . 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 260	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → 𝑦 ∈ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)))
69	68	ex 417	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (¬ (𝐺‘𝑦) = 𝑍 → 𝑦 ∈ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍))))
70	69	con1d 146	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (¬ 𝑦 ∈ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)) → (𝐺‘𝑦) = 𝑍))
71	70	impr 459	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)))) → (𝐺‘𝑦) = 𝑍)
72	22, 71	sylan2b 605	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∖ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)))) → (𝐺‘𝑦) = 𝑍)
73	21, 72	suppss 8186	. 2 ⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)))
74		suppssfifsupp 9336	. 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 1405	1 ⊢ (𝜑 → 𝐺 finSupp 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089  Vcvv 3455   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  ⟨cop 4595   class class class wbr 5109   ↦ cmpt 5192   × cxp 5659  dom cdm 5661   ↾ cres 5663   “ cima 5664  Fun wfun 6530   Fn wfn 6531  –onto→wfo 6534  ‘cfv 6536  (class class class)co 7410  2^nd c2nd 7981   supp csupp 8152  Fincfn 8939   finSupp cfsupp 9317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-2nd 7983  df-supp 8153  df-1o 8449  df-en 8940  df-dom 8941  df-fin 8943  df-fsupp 9318

This theorem is referenced by:  fedgmullem2  34020