Theorem fsuppcurry2 32877

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 7399	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥𝐹𝐶) = (𝑦𝐹𝐶))
3	2	cbvmptv 5203	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶)) = (𝑦 ∈ 𝐴 ↦ (𝑦𝐹𝐶))
4	1, 3	eqtri 2784	. . 3 ⊢ 𝐺 = (𝑦 ∈ 𝐴 ↦ (𝑦𝐹𝐶))
5		fsuppcurry2.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6	5	mptexd 7204	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ (𝑦𝐹𝐶)) ∈ V)
7	4, 6	eqeltrid 2865	. 2 ⊢ (𝜑 → 𝐺 ∈ V)
8	1	funmpt2 6556	. . 3 ⊢ Fun 𝐺
9	8	a1i 11	. 2 ⊢ (𝜑 → Fun 𝐺)
10		fsuppcurry2.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
11		fo1st 7986	. . . . 5 ⊢ 1st :V–onto→V
12		fofun 6775	. . . . 5 ⊢ (1st :V–onto→V → Fun 1st )
13	11, 12	ax-mp 5	. . . 4 ⊢ Fun 1st
14		funres 6559	. . . 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 9312	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
18		imafi 9255	. . 3 ⊢ ((Fun (1st ↾ (V × V)) ∧ (𝐹 supp 𝑍) ∈ Fin) → ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) ∈ Fin)
19	15, 17, 18	syl2anc 593	. 2 ⊢ (𝜑 → ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) ∈ Fin)
20		ovexd 7427	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑦𝐹𝐶) ∈ V)
21	20, 4	fmptd 7091	. . 3 ⊢ (𝜑 → 𝐺:𝐴⟶V)
22		eldif 3914	. . . 4 ⊢ (𝑦 ∈ (𝐴 ∖ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍))) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍))))
23		simplr 778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → 𝑦 ∈ 𝐴)
24		fsuppcurry2.c	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
25	24	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → 𝐶 ∈ 𝐵)
26	23, 25	opelxpd 5684	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵))
27		df-ov 7395	. . . . . . . . . . 11 ⊢ (𝑦𝐹𝐶) = (𝐹‘⟨𝑦, 𝐶⟩)
28		ovexd 7427	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝑦𝐹𝐶) ∈ V)
29	1, 2, 23, 28	fvmptd3 6995	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝐺‘𝑦) = (𝑦𝐹𝐶))
30		simpr 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ¬ (𝐺‘𝑦) = 𝑍)
31	30	neqned 2963	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝐺‘𝑦) ≠ 𝑍)
32	29, 31	eqnetrrd 3024	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → (𝑦𝐹𝐶) ≠ 𝑍)
33	27, 32	eqnetrrid 3031	. . . . . . . . . 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 7730	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
37		elsuppfn 8145	. . . . . . . . . . . 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 1389	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨𝑦, 𝐶⟩ ∈ (𝐹 supp 𝑍) ↔ (⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵) ∧ (𝐹‘⟨𝑦, 𝐶⟩) ≠ 𝑍)))
39	38	ad2antrr 736	. . . . . . . . . 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 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ⟨𝑦, 𝐶⟩ ∈ (𝐹 supp 𝑍))
41		simpr 488	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → 𝑧 = ⟨𝑦, 𝐶⟩)
42	41	fveq2d 6867	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ((1st ↾ (V × V))‘𝑧) = ((1st ↾ (V × V))‘⟨𝑦, 𝐶⟩))
43		xpss 5661	. . . . . . . . . . . 12 ⊢ (𝐴 × 𝐵) ⊆ (V × V)
44	26	adantr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵))
45	43, 44	sselid 3934	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ⟨𝑦, 𝐶⟩ ∈ (V × V))
46	45	fvresd 6883	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ((1st ↾ (V × V))‘⟨𝑦, 𝐶⟩) = (1st ‘⟨𝑦, 𝐶⟩))
47	23	adantr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → 𝑦 ∈ 𝐴)
48	25	adantr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → 𝐶 ∈ 𝐵)
49		op1stg 7978	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (1st ‘⟨𝑦, 𝐶⟩) = 𝑦)
50	47, 48, 49	syl2anc 593	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → (1st ‘⟨𝑦, 𝐶⟩) = 𝑦)
51	42, 46, 50	3eqtrd 2800	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ((1st ↾ (V × V))‘𝑧) = 𝑦)
52	40, 51	rspcedeq1vd 3588	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → ∃𝑧 ∈ (𝐹 supp 𝑍)((1st ↾ (V × V))‘𝑧) = 𝑦)
53		fofn 6776	. . . . . . . . . . . . 13 ⊢ (1st :V–onto→V → 1st Fn V)
54		fnresin 32776	. . . . . . . . . . . . 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 3960	. . . . . . . . . . . . . 14 ⊢ (V × V) ⊆ V
57		sseqin2 4175	. . . . . . . . . . . . . 14 ⊢ ((V × V) ⊆ V ↔ (V ∩ (V × V)) = (V × V))
58	56, 57	mpbi 232	. . . . . . . . . . . . 13 ⊢ (V ∩ (V × V)) = (V × V)
59	58	fneq2i 6615	. . . . . . . . . . . 12 ⊢ ((1st ↾ (V × V)) Fn (V ∩ (V × V)) ↔ (1st ↾ (V × V)) Fn (V × V))
60	55, 59	mpbi 232	. . . . . . . . . . 11 ⊢ (1st ↾ (V × V)) Fn (V × V)
61	60	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (1st ↾ (V × V)) Fn (V × V))
62		suppssdm 8152	. . . . . . . . . . . 12 ⊢ (𝐹 supp 𝑍) ⊆ dom 𝐹
63	34	fndmd 6622	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐹 = (𝐴 × 𝐵))
64	62, 63	sseqtrid 3978	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐴 × 𝐵))
65	64, 43	sstrdi 3948	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (V × V))
66	61, 65	fvelimabd 6936	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) ↔ ∃𝑧 ∈ (𝐹 supp 𝑍)((1st ↾ (V × V))‘𝑧) = 𝑦))
67	66	ad2antrr 736	. . . . . . . 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 259	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ (𝐺‘𝑦) = 𝑍) → 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))
69	68	ex 416	. . . . . 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 458	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))) → (𝐺‘𝑦) = 𝑍)
72	22, 71	sylan2b 603	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))) → (𝐺‘𝑦) = 𝑍)
73	21, 72	suppss 8169	. 2 ⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))
74		suppssfifsupp 9323	. 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 1396	1 ⊢ (𝜑 → 𝐺 finSupp 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∃wrex 3085  Vcvv 3453   ∖ cdif 3901   ∩ cin 3903   ⊆ wss 3904  ⟨cop 4587   class class class wbr 5099   ↦ cmpt 5180   × cxp 5643  dom cdm 5645   ↾ cres 5647   “ cima 5648  Fun wfun 6511   Fn wfn 6512  –onto→wfo 6515  ‘cfv 6517  (class class class)co 7392  1^st c1st 7964   supp csupp 8135  Fincfn 8923   finSupp cfsupp 9304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-supp 8136  df-1o 8432  df-en 8924  df-dom 8925  df-fin 8927  df-fsupp 9305

This theorem is referenced by: (None)