Theorem evlselvlem 43170

Description: Lemma for evlselv 43171. 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 4437	. . . . . 6 ⊢ (𝐽 ⊆ 𝐼 ↔ ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
4	2, 3	sylib 220	. . . . 5 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
5	4	adantr 484	. . . 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 21970	. . . . . 6 ⊢ (𝑐 ∈ 𝐶 → 𝑐:(𝐼 ∖ 𝐽)⟶ℕ0)
8	7	ad2antrl 738	. . . . 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 21970	. . . . . 6 ⊢ (𝑒 ∈ 𝐸 → 𝑒:𝐽⟶ℕ0)
11	10	ad2antll 739	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑒:𝐽⟶ℕ0)
12		disjdifr 4427	. . . . . 6 ⊢ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅
13	12	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅)
14	8, 11, 13	fun2d 6728	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒):((𝐼 ∖ 𝐽) ∪ 𝐽)⟶ℕ0)
15	5, 14	feq2dd 6677	. . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒):𝐼⟶ℕ0)
16		unexg 7726	. . . . . 6 ⊢ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) → (𝑐 ∪ 𝑒) ∈ V)
17	16	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒) ∈ V)
18		0zd 12580	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 0 ∈ ℤ)
19	14	ffund 6696	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → Fun (𝑐 ∪ 𝑒))
20	6	psrbagfsupp 21971	. . . . . . 7 ⊢ (𝑐 ∈ 𝐶 → 𝑐 finSupp 0)
21	20	ad2antrl 738	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑐 finSupp 0)
22	9	psrbagfsupp 21971	. . . . . . 7 ⊢ (𝑒 ∈ 𝐸 → 𝑒 finSupp 0)
23	22	ad2antll 739	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑒 finSupp 0)
24	21, 23	fsuppun 9333	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) supp 0) ∈ Fin)
25	17, 18, 19, 24	isfsuppd 9312	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒) finSupp 0)
26		fcdmnn0fsuppg 12541	. . . . 5 ⊢ (((𝑐 ∪ 𝑒) ∈ V ∧ (𝑐 ∪ 𝑒):((𝐼 ∖ 𝐽) ∪ 𝐽)⟶ℕ0) → ((𝑐 ∪ 𝑒) finSupp 0 ↔ (◡(𝑐 ∪ 𝑒) “ ℕ) ∈ Fin))
27	17, 14, 26	syl2anc 593	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) finSupp 0 ↔ (◡(𝑐 ∪ 𝑒) “ ℕ) ∈ Fin))
28	25, 27	mpbid 234	. . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (◡(𝑐 ∪ 𝑒) “ ℕ) ∈ Fin)
29		evlselvlem.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
30	29	adantr 484	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝐼 ∈ 𝑉)
31		evlselvlem.d	. . . . 5 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
32	31	psrbag 21969	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ((𝑐 ∪ 𝑒) ∈ 𝐷 ↔ ((𝑐 ∪ 𝑒):𝐼⟶ℕ0 ∧ (◡(𝑐 ∪ 𝑒) “ ℕ) ∈ Fin)))
33	30, 32	syl 17	. . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) ∈ 𝐷 ↔ ((𝑐 ∪ 𝑒):𝐼⟶ℕ0 ∧ (◡(𝑐 ∪ 𝑒) “ ℕ) ∈ Fin)))
34	15, 28, 33	mpbir2and 723	. 2 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒) ∈ 𝐷)
35	29	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝐼 ∈ 𝑉)
36		difssd 4090	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (𝐼 ∖ 𝐽) ⊆ 𝐼)
37		simpr 488	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ 𝐷)
38	31, 6, 35, 36, 37	psrbagres 21982	. 2 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (𝑑 ↾ (𝐼 ∖ 𝐽)) ∈ 𝐶)
39	2	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝐽 ⊆ 𝐼)
40	31, 9, 35, 39, 37	psrbagres 21982	. 2 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (𝑑 ↾ 𝐽) ∈ 𝐸)
41	31	psrbagf 21970	. . . . . . . 8 ⊢ (𝑑 ∈ 𝐷 → 𝑑:𝐼⟶ℕ0)
42	41	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑑:𝐼⟶ℕ0)
43	42	freld 6698	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → Rel 𝑑)
44	42	fdmd 6702	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → dom 𝑑 = 𝐼)
45	39, 3	sylib 220	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
46	44, 45	eqtr4d 2800	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → dom 𝑑 = ((𝐼 ∖ 𝐽) ∪ 𝐽))
47		reldmun 6020	. . . . . 6 ⊢ ((Rel 𝑑 ∧ dom 𝑑 = ((𝐼 ∖ 𝐽) ∪ 𝐽)) → 𝑑 = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽)))
48	43, 46, 47	syl2anc 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑑 = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽)))
49	48	adantrl 726	. . . 4 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) ∧ 𝑑 ∈ 𝐷)) → 𝑑 = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽)))
50		uneq12 4116	. . . . 5 ⊢ ((𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽)) → (𝑐 ∪ 𝑒) = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽)))
51	50	eqeq2d 2773	. . . 4 ⊢ ((𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽)) → (𝑑 = (𝑐 ∪ 𝑒) ↔ 𝑑 = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽))))
52	49, 51	syl5ibrcom 249	. . 3 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) ∧ 𝑑 ∈ 𝐷)) → ((𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽)) → 𝑑 = (𝑐 ∪ 𝑒)))
53	8	ffnd 6692	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑐 Fn (𝐼 ∖ 𝐽))
54	11	ffnd 6692	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑒 Fn 𝐽)
55		fnunres1 6633	. . . . . . . 8 ⊢ ((𝑐 Fn (𝐼 ∖ 𝐽) ∧ 𝑒 Fn 𝐽 ∧ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅) → ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) = 𝑐)
56	53, 54, 13, 55	syl3anc 1390	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) = 𝑐)
57	56	eqcomd 2768	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)))
58		fnunres2 6634	. . . . . . . 8 ⊢ ((𝑐 Fn (𝐼 ∖ 𝐽) ∧ 𝑒 Fn 𝐽 ∧ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅) → ((𝑐 ∪ 𝑒) ↾ 𝐽) = 𝑒)
59	53, 54, 13, 58	syl3anc 1390	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) ↾ 𝐽) = 𝑒)
60	59	eqcomd 2768	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽))
61	57, 60	jca 519	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽)))
62	61	adantrr 727	. . . 4 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) ∧ 𝑑 ∈ 𝐷)) → (𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽)))
63		reseq1 5959	. . . . . 6 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → (𝑑 ↾ (𝐼 ∖ 𝐽)) = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)))
64	63	eqeq2d 2773	. . . . 5 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → (𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ↔ 𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽))))
65		reseq1 5959	. . . . . 6 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → (𝑑 ↾ 𝐽) = ((𝑐 ∪ 𝑒) ↾ 𝐽))
66	65	eqeq2d 2773	. . . . 5 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → (𝑒 = (𝑑 ↾ 𝐽) ↔ 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽)))
67	64, 66	anbi12d 641	. . . 4 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → ((𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽)) ↔ (𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽))))
68	62, 67	syl5ibrcom 249	. . 3 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) ∧ 𝑑 ∈ 𝐷)) → (𝑑 = (𝑐 ∪ 𝑒) → (𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽))))
69	52, 68	impbid 214	. 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 8105	1 ⊢ (𝜑 → 𝐻:(𝐶 × 𝐸)–1-1-onto→𝐷)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  {crab 3414  Vcvv 3454   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285   class class class wbr 5100   × cxp 5645  ^◡ccnv 5646  dom cdm 5647   ↾ cres 5649   “ cima 5650  Rel wrel 5652   Fn wfn 6516  ⟶wf 6517  –1-1-onto→wf1o 6520  (class class class)co 7396   ∈ cmpo 7398   ↑_m cmap 8808  Fincfn 8927   finSupp cfsupp 9307  0cc0 11073  ℕcn 12210  ℕ₀cn0 12481  ℤcz 12568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  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 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-supp 8141  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-er 8678  df-map 8810  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fsupp 9308  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-neg 11417  df-nn 12211  df-n0 12482  df-z 12569

This theorem is referenced by:  evlselv  43171