Theorem evlselvlem 43212

Description: Lemma for evlselv 43213. Used to re-index to and from bags of variables in 𝐼 and bags of variables in the subsets 𝐽 and 𝐼 ∖ 𝐽. (Contributed by SN, 10-Mar-2025.)

Hypotheses

Ref	Expression
evlselvlem.d	⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
evlselvlem.e	⊢ 𝐸 = {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}
evlselvlem.c	⊢ 𝐶 = {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}
evlselvlem.h	⊢ 𝐻 = (𝑐 ∈ 𝐶, 𝑒 ∈ 𝐸 ↦ (𝑐 ∪ 𝑒))
evlselvlem.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
evlselvlem.j	⊢ (𝜑 → 𝐽 ⊆ 𝐼)

Assertion

Ref	Expression
evlselvlem	⊢ (𝜑 → 𝐻:(𝐶 × 𝐸)–1-1-onto→𝐷)

Distinct variable groups:   𝑓,𝑐,𝐼   𝑓,𝐽   𝐼,𝑐,𝑒,ℎ   𝐽,𝑐,𝑒,𝑔   𝐶,𝑐,𝑒   𝐷,𝑐,𝑒   𝐸,𝑐,𝑒   𝜑,𝑐,𝑒

Allowed substitution hints:   𝜑(𝑓,𝑔,ℎ)   𝐶(𝑓,𝑔,ℎ)   𝐷(𝑓,𝑔,ℎ)   𝐸(𝑓,𝑔,ℎ)   𝐻(𝑒,𝑓,𝑔,ℎ,𝑐)   𝐼(𝑔)   𝐽(ℎ)   𝑉(𝑒,𝑓,𝑔,ℎ,𝑐)

Proof of Theorem evlselvlem

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		evlselvlem.h	. 2 ⊢ 𝐻 = (𝑐 ∈ 𝐶, 𝑒 ∈ 𝐸 ↦ (𝑐 ∪ 𝑒))
2		evlselvlem.j	. . . . . 6 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
3		undifr 4449	. . . . . 6 ⊢ (𝐽 ⊆ 𝐼 ↔ ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
4	2, 3	sylib 221	. . . . 5 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
5	4	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
6		evlselvlem.c	. . . . . . 7 ⊢ 𝐶 = {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}
7	6	psrbagf 22037	. . . . . 6 ⊢ (𝑐 ∈ 𝐶 → 𝑐:(𝐼 ∖ 𝐽)⟶ℕ0)
8	7	ad2antrl 740	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑐:(𝐼 ∖ 𝐽)⟶ℕ0)
9		evlselvlem.e	. . . . . . 7 ⊢ 𝐸 = {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}
10	9	psrbagf 22037	. . . . . 6 ⊢ (𝑒 ∈ 𝐸 → 𝑒:𝐽⟶ℕ0)
11	10	ad2antll 741	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑒:𝐽⟶ℕ0)
12		disjdifr 4439	. . . . . 6 ⊢ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅
13	12	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅)
14	8, 11, 13	fun2d 6743	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒):((𝐼 ∖ 𝐽) ∪ 𝐽)⟶ℕ0)
15	5, 14	feq2dd 6692	. . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒):𝐼⟶ℕ0)
16		unexg 7742	. . . . . 6 ⊢ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) → (𝑐 ∪ 𝑒) ∈ V)
17	16	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒) ∈ V)
18		0zd 12603	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 0 ∈ ℤ)
19	14	ffund 6711	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → Fun (𝑐 ∪ 𝑒))
20	6	psrbagfsupp 22038	. . . . . . 7 ⊢ (𝑐 ∈ 𝐶 → 𝑐 finSupp 0)
21	20	ad2antrl 740	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑐 finSupp 0)
22	9	psrbagfsupp 22038	. . . . . . 7 ⊢ (𝑒 ∈ 𝐸 → 𝑒 finSupp 0)
23	22	ad2antll 741	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑒 finSupp 0)
24	21, 23	fsuppun 9347	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) supp 0) ∈ Fin)
25	17, 18, 19, 24	isfsuppd 9326	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒) finSupp 0)
26		fcdmnn0fsuppg 12564	. . . . 5 ⊢ (((𝑐 ∪ 𝑒) ∈ V ∧ (𝑐 ∪ 𝑒):((𝐼 ∖ 𝐽) ∪ 𝐽)⟶ℕ0) → ((𝑐 ∪ 𝑒) finSupp 0 ↔ (◡(𝑐 ∪ 𝑒) “ ℕ) ∈ Fin))
27	17, 14, 26	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) finSupp 0 ↔ (◡(𝑐 ∪ 𝑒) “ ℕ) ∈ Fin))
28	25, 27	mpbid 235	. . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (◡(𝑐 ∪ 𝑒) “ ℕ) ∈ Fin)
29		evlselvlem.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
30	29	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝐼 ∈ 𝑉)
31		evlselvlem.d	. . . . 5 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
32	31	psrbag 22036	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ((𝑐 ∪ 𝑒) ∈ 𝐷 ↔ ((𝑐 ∪ 𝑒):𝐼⟶ℕ0 ∧ (◡(𝑐 ∪ 𝑒) “ ℕ) ∈ Fin)))
33	30, 32	syl 18	. . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) ∈ 𝐷 ↔ ((𝑐 ∪ 𝑒):𝐼⟶ℕ0 ∧ (◡(𝑐 ∪ 𝑒) “ ℕ) ∈ Fin)))
34	15, 28, 33	mpbir2and 725	. 2 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒) ∈ 𝐷)
35	29	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝐼 ∈ 𝑉)
36		difssd 4099	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (𝐼 ∖ 𝐽) ⊆ 𝐼)
37		simpr 489	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ 𝐷)
38	31, 6, 35, 36, 37	psrbagres 22049	. 2 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (𝑑 ↾ (𝐼 ∖ 𝐽)) ∈ 𝐶)
39	2	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝐽 ⊆ 𝐼)
40	31, 9, 35, 39, 37	psrbagres 22049	. 2 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (𝑑 ↾ 𝐽) ∈ 𝐸)
41	31	psrbagf 22037	. . . . . . . 8 ⊢ (𝑑 ∈ 𝐷 → 𝑑:𝐼⟶ℕ0)
42	41	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑑:𝐼⟶ℕ0)
43	42	freld 6713	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → Rel 𝑑)
44	42	fdmd 6717	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → dom 𝑑 = 𝐼)
45	39, 3	sylib 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
46	44, 45	eqtr4d 2807	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → dom 𝑑 = ((𝐼 ∖ 𝐽) ∪ 𝐽))
47		reldmun 6034	. . . . . 6 ⊢ ((Rel 𝑑 ∧ dom 𝑑 = ((𝐼 ∖ 𝐽) ∪ 𝐽)) → 𝑑 = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽)))
48	43, 46, 47	syl2anc 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑑 = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽)))
49	48	adantrl 728	. . . 4 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) ∧ 𝑑 ∈ 𝐷)) → 𝑑 = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽)))
50		uneq12 4125	. . . . 5 ⊢ ((𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽)) → (𝑐 ∪ 𝑒) = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽)))
51	50	eqeq2d 2780	. . . 4 ⊢ ((𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽)) → (𝑑 = (𝑐 ∪ 𝑒) ↔ 𝑑 = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽))))
52	49, 51	syl5ibrcom 250	. . 3 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) ∧ 𝑑 ∈ 𝐷)) → ((𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽)) → 𝑑 = (𝑐 ∪ 𝑒)))
53	8	ffnd 6707	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑐 Fn (𝐼 ∖ 𝐽))
54	11	ffnd 6707	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑒 Fn 𝐽)
55		fnunres1 6648	. . . . . . . 8 ⊢ ((𝑐 Fn (𝐼 ∖ 𝐽) ∧ 𝑒 Fn 𝐽 ∧ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅) → ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) = 𝑐)
56	53, 54, 13, 55	syl3anc 1396	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) = 𝑐)
57	56	eqcomd 2775	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)))
58		fnunres2 6649	. . . . . . . 8 ⊢ ((𝑐 Fn (𝐼 ∖ 𝐽) ∧ 𝑒 Fn 𝐽 ∧ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅) → ((𝑐 ∪ 𝑒) ↾ 𝐽) = 𝑒)
59	53, 54, 13, 58	syl3anc 1396	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) ↾ 𝐽) = 𝑒)
60	59	eqcomd 2775	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽))
61	57, 60	jca 520	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽)))
62	61	adantrr 729	. . . 4 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) ∧ 𝑑 ∈ 𝐷)) → (𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽)))
63		reseq1 5973	. . . . . 6 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → (𝑑 ↾ (𝐼 ∖ 𝐽)) = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)))
64	63	eqeq2d 2780	. . . . 5 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → (𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ↔ 𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽))))
65		reseq1 5973	. . . . . 6 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → (𝑑 ↾ 𝐽) = ((𝑐 ∪ 𝑒) ↾ 𝐽))
66	65	eqeq2d 2780	. . . . 5 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → (𝑒 = (𝑑 ↾ 𝐽) ↔ 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽)))
67	64, 66	anbi12d 643	. . . 4 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → ((𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽)) ↔ (𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽))))
68	62, 67	syl5ibrcom 250	. . 3 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) ∧ 𝑑 ∈ 𝐷)) → (𝑑 = (𝑐 ∪ 𝑒) → (𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽))))
69	52, 68	impbid 215	. 2 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) ∧ 𝑑 ∈ 𝐷)) → ((𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽)) ↔ 𝑑 = (𝑐 ∪ 𝑒)))
70	1, 34, 38, 40, 69	mpof1o2d 8121	1 ⊢ (𝜑 → 𝐻:(𝐶 × 𝐸)–1-1-onto→𝐷)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {crab 3423  Vcvv 3463   ∖ cdif 3910   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294   class class class wbr 5113   × cxp 5660  ^◡ccnv 5661  dom cdm 5662   ↾ cres 5664   “ cima 5665  Rel wrel 5667   Fn wfn 6532  ⟶wf 6533  –1-1-onto→wf1o 6536  (class class class)co 7411   ∈ cmpo 7413   ↑_m cmap 8824  Fincfn 8943   finSupp cfsupp 9321  0cc0 11100  ℕcn 12233  ℕ₀cn0 12504  ℤcz 12591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8157  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fsupp 9322  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-neg 11444  df-nn 12234  df-n0 12505  df-z 12592

This theorem is referenced by:  evlselv  43213