Theorem fsuppcurry1 31821

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 7401	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐶𝐹𝑥) = (𝐶𝐹𝑦))
3	2	cbvmptv 5254	. . . 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 7210	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ (𝐶𝐹𝑦)) ∈ V)
7	4, 6	eqeltrid 2836	. 2 ⊢ (𝜑 → 𝐺 ∈ V)
8	1	funmpt2 6576	. . 3 ⊢ Fun 𝐺
9	8	a1i 11	. 2 ⊢ (𝜑 → Fun 𝐺)
10		fsuppcurry1.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
11		fo2nd 7978	. . . . 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 6579	. . . 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 9352	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
18		imafi 9158	. . 3 ⊢ ((Fun (2nd ↾ (V × V)) ∧ (𝐹 supp 𝑍) ∈ Fin) → ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)) ∈ Fin)
19	15, 17, 18	syl2anc 584	. 2 ⊢ (𝜑 → ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)) ∈ Fin)
20		ovexd 7428	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐶𝐹𝑦) ∈ V)
21	20, 4	fmptd 7098	. . 3 ⊢ (𝜑 → 𝐺:𝐵⟶V)
22		eldif 3954	. . . 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 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → 𝐶 ∈ 𝐴)
25		simplr 767	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → 𝑦 ∈ 𝐵)
26	24, 25	opelxpd 5707	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ⟨𝐶, 𝑦⟩ ∈ (𝐴 × 𝐵))
27		df-ov 7396	. . . . . . . . . . 11 ⊢ (𝐶𝐹𝑦) = (𝐹‘⟨𝐶, 𝑦⟩)
28		ovexd 7428	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝐶𝐹𝑦) ∈ V)
29	1, 2, 25, 28	fvmptd3 7007	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝐺‘𝑦) = (𝐶𝐹𝑦))
30		simpr 485	. . . . . . . . . . . . 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 7721	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
37		elsuppfn 8138	. . . . . . . . . . . 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 1371	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨𝐶, 𝑦⟩ ∈ (𝐹 supp 𝑍) ↔ (⟨𝐶, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ (𝐹‘⟨𝐶, 𝑦⟩) ≠ 𝑍)))
39	38	ad2antrr 724	. . . . . . . . . 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 711	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ⟨𝐶, 𝑦⟩ ∈ (𝐹 supp 𝑍))
41		simpr 485	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → 𝑧 = ⟨𝐶, 𝑦⟩)
42	41	fveq2d 6882	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → ((2nd ↾ (V × V))‘𝑧) = ((2nd ↾ (V × V))‘⟨𝐶, 𝑦⟩))
43		xpss 5685	. . . . . . . . . . . 12 ⊢ (𝐴 × 𝐵) ⊆ (V × V)
44	26	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → ⟨𝐶, 𝑦⟩ ∈ (𝐴 × 𝐵))
45	43, 44	sselid 3976	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → ⟨𝐶, 𝑦⟩ ∈ (V × V))
46	45	fvresd 6898	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → ((2nd ↾ (V × V))‘⟨𝐶, 𝑦⟩) = (2nd ‘⟨𝐶, 𝑦⟩))
47	24	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → 𝐶 ∈ 𝐴)
48	25	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝐶, 𝑦⟩) → 𝑦 ∈ 𝐵)
49		op2ndg 7970	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (2nd ‘⟨𝐶, 𝑦⟩) = 𝑦)
50	47, 48, 49	syl2anc 584	. . . . . . . . . 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 3614	. . . . . . . 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 31718	. . . . . . . . . . . . 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 4002	. . . . . . . . . . . . . 14 ⊢ (V × V) ⊆ V
57		sseqin2 4211	. . . . . . . . . . . . . 14 ⊢ ((V × V) ⊆ V ↔ (V ∩ (V × V)) = (V × V))
58	56, 57	mpbi 229	. . . . . . . . . . . . 13 ⊢ (V ∩ (V × V)) = (V × V)
59	58	fneq2i 6636	. . . . . . . . . . . 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 8144	. . . . . . . . . . . 12 ⊢ (𝐹 supp 𝑍) ⊆ dom 𝐹
63	34	fndmd 6643	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐹 = (𝐴 × 𝐵))
64	62, 63	sseqtrid 4030	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐴 × 𝐵))
65	64, 43	sstrdi 3990	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (V × V))
66	61, 65	fvelimabd 6951	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)) ↔ ∃𝑧 ∈ (𝐹 supp 𝑍)((2nd ↾ (V × V))‘𝑧) = 𝑦))
67	66	ad2antrr 724	. . . . . . . 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 413	. . . . . 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 455	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)))) → (𝐺‘𝑦) = 𝑍)
72	22, 71	sylan2b 594	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∖ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)))) → (𝐺‘𝑦) = 𝑍)
73	21, 72	suppss 8161	. 2 ⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ ((2nd ↾ (V × V)) “ (𝐹 supp 𝑍)))
74		suppssfifsupp 9361	. 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 1378	1 ⊢ (𝜑 → 𝐺 finSupp 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2939  ∃wrex 3069  Vcvv 3473   ∖ cdif 3941   ∩ cin 3943   ⊆ wss 3944  ⟨cop 4628   class class class wbr 5141   ↦ cmpt 5224   × cxp 5667  dom cdm 5669   ↾ cres 5671   “ cima 5672  Fun wfun 6526   Fn wfn 6527  –onto→wfo 6530  ‘cfv 6532  (class class class)co 7393  2^nd c2nd 7956   supp csupp 8128  Fincfn 8922   finSupp cfsupp 9344

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  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 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-2nd 7958  df-supp 8129  df-1o 8448  df-en 8923  df-fin 8926  df-fsupp 9345

This theorem is referenced by:  fedgmullem2  32553