Theorem evlselvlem 43435

Description: Lemma for evlselv 43436. 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 4439	. . . . . 6 ⊢ (𝐽 ⊆ 𝐼 ↔ ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
4	2, 3	sylib 221	. . . . 5 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
5	4	adantr 486	. . . 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 22134	. . . . . 6 ⊢ (𝑐 ∈ 𝐶 → 𝑐:(𝐼 ∖ 𝐽)⟶ℕ0)
8	7	ad2antrl 741	. . . . 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 22134	. . . . . 6 ⊢ (𝑒 ∈ 𝐸 → 𝑒:𝐽⟶ℕ0)
11	10	ad2antll 742	. . . . 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 6740	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒):((𝐼 ∖ 𝐽) ∪ 𝐽)⟶ℕ0)
15	5, 14	feq2dd 6689	. . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒):𝐼⟶ℕ0)
16		unexg 7746	. . . . . 6 ⊢ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) → (𝑐 ∪ 𝑒) ∈ V)
17	16	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒) ∈ V)
18		0zd 12628	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 0 ∈ ℤ)
19	14	ffund 6708	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → Fun (𝑐 ∪ 𝑒))
20	6	psrbagfsupp 22135	. . . . . . 7 ⊢ (𝑐 ∈ 𝐶 → 𝑐 finSupp 0)
21	20	ad2antrl 741	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑐 finSupp 0)
22	9	psrbagfsupp 22135	. . . . . . 7 ⊢ (𝑒 ∈ 𝐸 → 𝑒 finSupp 0)
23	22	ad2antll 742	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑒 finSupp 0)
24	21, 23	fsuppun 9358	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) supp 0) ∈ Fin)
25	17, 18, 19, 24	isfsuppd 9337	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒) finSupp 0)
26		fcdmnn0fsuppg 12589	. . . . 5 ⊢ (((𝑐 ∪ 𝑒) ∈ V ∧ (𝑐 ∪ 𝑒):((𝐼 ∖ 𝐽) ∪ 𝐽)⟶ℕ0) → ((𝑐 ∪ 𝑒) finSupp 0 ↔ (◡(𝑐 ∪ 𝑒) “ ℕ) ∈ Fin))
27	17, 14, 26	syl2anc 596	. . . 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 486	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝐼 ∈ 𝑉)
31		evlselvlem.d	. . . . 5 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
32	31	psrbag 22133	. . . 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 726	. 2 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒) ∈ 𝐷)
35	29	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝐼 ∈ 𝑉)
36		difssd 4084	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (𝐼 ∖ 𝐽) ⊆ 𝐼)
37		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ 𝐷)
38	31, 6, 35, 36, 37	psrbagres 22146	. 2 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (𝑑 ↾ (𝐼 ∖ 𝐽)) ∈ 𝐶)
39	2	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝐽 ⊆ 𝐼)
40	31, 9, 35, 39, 37	psrbagres 22146	. 2 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (𝑑 ↾ 𝐽) ∈ 𝐸)
41	31	psrbagf 22134	. . . . . . . 8 ⊢ (𝑑 ∈ 𝐷 → 𝑑:𝐼⟶ℕ0)
42	41	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑑:𝐼⟶ℕ0)
43	42	freld 6710	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → Rel 𝑑)
44	42	fdmd 6714	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → dom 𝑑 = 𝐼)
45	39, 3	sylib 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
46	44, 45	eqtr4d 2798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → dom 𝑑 = ((𝐼 ∖ 𝐽) ∪ 𝐽))
47		reldmun 6027	. . . . . 6 ⊢ ((Rel 𝑑 ∧ dom 𝑑 = ((𝐼 ∖ 𝐽) ∪ 𝐽)) → 𝑑 = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽)))
48	43, 46, 47	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑑 = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽)))
49	48	adantrl 729	. . . 4 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) ∧ 𝑑 ∈ 𝐷)) → 𝑑 = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽)))
50		uneq12 4110	. . . . 5 ⊢ ((𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽)) → (𝑐 ∪ 𝑒) = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽)))
51	50	eqeq2d 2771	. . . 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 6704	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑐 Fn (𝐼 ∖ 𝐽))
54	11	ffnd 6704	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑒 Fn 𝐽)
55		fnunres1 6645	. . . . . . . 8 ⊢ ((𝑐 Fn (𝐼 ∖ 𝐽) ∧ 𝑒 Fn 𝐽 ∧ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅) → ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) = 𝑐)
56	53, 54, 13, 55	syl3anc 1398	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) = 𝑐)
57	56	eqcomd 2766	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)))
58		fnunres2 6646	. . . . . . . 8 ⊢ ((𝑐 Fn (𝐼 ∖ 𝐽) ∧ 𝑒 Fn 𝐽 ∧ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅) → ((𝑐 ∪ 𝑒) ↾ 𝐽) = 𝑒)
59	53, 54, 13, 58	syl3anc 1398	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) ↾ 𝐽) = 𝑒)
60	59	eqcomd 2766	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽))
61	57, 60	jca 521	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽)))
62	61	adantrr 730	. . . 4 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) ∧ 𝑑 ∈ 𝐷)) → (𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽)))
63		reseq1 5966	. . . . . 6 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → (𝑑 ↾ (𝐼 ∖ 𝐽)) = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)))
64	63	eqeq2d 2771	. . . . 5 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → (𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ↔ 𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽))))
65		reseq1 5966	. . . . . 6 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → (𝑑 ↾ 𝐽) = ((𝑐 ∪ 𝑒) ↾ 𝐽))
66	65	eqeq2d 2771	. . . . 5 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → (𝑒 = (𝑑 ↾ 𝐽) ↔ 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽)))
67	64, 66	anbi12d 644	. . . 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 8124	1 ⊢ (𝜑 → 𝐻:(𝐶 × 𝐸)–1-1-onto→𝐷)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3412  Vcvv 3450   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279   class class class wbr 5103   × cxp 5653  ^◡ccnv 5654  dom cdm 5655   ↾ cres 5657   “ cima 5658  Rel wrel 5660   Fn wfn 6528  ⟶wf 6529  –1-1-onto→wf1o 6532  (class class class)co 7414   ∈ cmpo 7416   ↑_m cmap 8827  Fincfn 8953   finSupp cfsupp 9332  0cc0 11125  ℕcn 12258  ℕ₀cn0 12529  ℤcz 12616

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11181  ax-resscn 11182  ax-1cn 11183  ax-icn 11184  ax-addcl 11185  ax-addrcl 11186  ax-mulcl 11187  ax-mulrcl 11188  ax-mulcom 11189  ax-addass 11190  ax-mulass 11191  ax-distr 11192  ax-i2m1 11193  ax-1ne0 11194  ax-1rid 11195  ax-rnegex 11196  ax-rrecex 11197  ax-cnre 11198  ax-pre-lttri 11199  ax-pre-lttrn 11200  ax-pre-ltadd 11201

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-supp 8160  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8456  df-er 8697  df-map 8829  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-fsupp 9333  df-pnf 11270  df-mnf 11271  df-xr 11272  df-ltxr 11273  df-le 11274  df-neg 11469  df-nn 12259  df-n0 12530  df-z 12617

This theorem is used by:  evlselv  43436