Theorem evlselvlem 42627

Description: Lemma for evlselv 42628. 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.c	. . . . . . 7 ⊢ 𝐶 = {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}
3	2	psrbagf 21855	. . . . . 6 ⊢ (𝑐 ∈ 𝐶 → 𝑐:(𝐼 ∖ 𝐽)⟶ℕ0)
4	3	ad2antrl 728	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑐:(𝐼 ∖ 𝐽)⟶ℕ0)
5		evlselvlem.e	. . . . . . 7 ⊢ 𝐸 = {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}
6	5	psrbagf 21855	. . . . . 6 ⊢ (𝑒 ∈ 𝐸 → 𝑒:𝐽⟶ℕ0)
7	6	ad2antll 729	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑒:𝐽⟶ℕ0)
8		disjdifr 4420	. . . . . 6 ⊢ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅
9	8	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅)
10	4, 7, 9	fun2d 6687	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒):((𝐼 ∖ 𝐽) ∪ 𝐽)⟶ℕ0)
11		evlselvlem.j	. . . . . . 7 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
12		undifr 4430	. . . . . . 7 ⊢ (𝐽 ⊆ 𝐼 ↔ ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
13	11, 12	sylib 218	. . . . . 6 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
14	13	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
15	14	feq2d 6635	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒):((𝐼 ∖ 𝐽) ∪ 𝐽)⟶ℕ0 ↔ (𝑐 ∪ 𝑒):𝐼⟶ℕ0))
16	10, 15	mpbid 232	. . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒):𝐼⟶ℕ0)
17		unexg 7676	. . . . . 6 ⊢ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) → (𝑐 ∪ 𝑒) ∈ V)
18	17	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒) ∈ V)
19		0zd 12480	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 0 ∈ ℤ)
20	10	ffund 6655	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → Fun (𝑐 ∪ 𝑒))
21	2	psrbagfsupp 21856	. . . . . . 7 ⊢ (𝑐 ∈ 𝐶 → 𝑐 finSupp 0)
22	21	ad2antrl 728	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑐 finSupp 0)
23	5	psrbagfsupp 21856	. . . . . . 7 ⊢ (𝑒 ∈ 𝐸 → 𝑒 finSupp 0)
24	23	ad2antll 729	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑒 finSupp 0)
25	22, 24	fsuppun 9271	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) supp 0) ∈ Fin)
26	18, 19, 20, 25	isfsuppd 9250	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒) finSupp 0)
27		fcdmnn0fsuppg 12441	. . . . 5 ⊢ (((𝑐 ∪ 𝑒) ∈ V ∧ (𝑐 ∪ 𝑒):((𝐼 ∖ 𝐽) ∪ 𝐽)⟶ℕ0) → ((𝑐 ∪ 𝑒) finSupp 0 ↔ (◡(𝑐 ∪ 𝑒) “ ℕ) ∈ Fin))
28	18, 10, 27	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) finSupp 0 ↔ (◡(𝑐 ∪ 𝑒) “ ℕ) ∈ Fin))
29	26, 28	mpbid 232	. . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (◡(𝑐 ∪ 𝑒) “ ℕ) ∈ Fin)
30		evlselvlem.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
31	30	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝐼 ∈ 𝑉)
32		evlselvlem.d	. . . . 5 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
33	32	psrbag 21854	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ((𝑐 ∪ 𝑒) ∈ 𝐷 ↔ ((𝑐 ∪ 𝑒):𝐼⟶ℕ0 ∧ (◡(𝑐 ∪ 𝑒) “ ℕ) ∈ Fin)))
34	31, 33	syl 17	. . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) ∈ 𝐷 ↔ ((𝑐 ∪ 𝑒):𝐼⟶ℕ0 ∧ (◡(𝑐 ∪ 𝑒) “ ℕ) ∈ Fin)))
35	16, 29, 34	mpbir2and 713	. 2 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒) ∈ 𝐷)
36	30	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝐼 ∈ 𝑉)
37		difssd 4084	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (𝐼 ∖ 𝐽) ⊆ 𝐼)
38		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ 𝐷)
39	32, 2, 36, 37, 38	psrbagres 42587	. 2 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (𝑑 ↾ (𝐼 ∖ 𝐽)) ∈ 𝐶)
40	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝐽 ⊆ 𝐼)
41	32, 5, 36, 40, 38	psrbagres 42587	. 2 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (𝑑 ↾ 𝐽) ∈ 𝐸)
42	32	psrbagf 21855	. . . . . . . 8 ⊢ (𝑑 ∈ 𝐷 → 𝑑:𝐼⟶ℕ0)
43	42	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑑:𝐼⟶ℕ0)
44	43	freld 6657	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → Rel 𝑑)
45	43	fdmd 6661	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → dom 𝑑 = 𝐼)
46	40, 12	sylib 218	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
47	45, 46	eqtr4d 2769	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → dom 𝑑 = ((𝐼 ∖ 𝐽) ∪ 𝐽))
48	8	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅)
49		reldisjun 5980	. . . . . 6 ⊢ ((Rel 𝑑 ∧ dom 𝑑 = ((𝐼 ∖ 𝐽) ∪ 𝐽) ∧ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅) → 𝑑 = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽)))
50	44, 47, 48, 49	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑑 = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽)))
51	50	adantrl 716	. . . 4 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) ∧ 𝑑 ∈ 𝐷)) → 𝑑 = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽)))
52		uneq12 4110	. . . . 5 ⊢ ((𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽)) → (𝑐 ∪ 𝑒) = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽)))
53	52	eqeq2d 2742	. . . 4 ⊢ ((𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽)) → (𝑑 = (𝑐 ∪ 𝑒) ↔ 𝑑 = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽))))
54	51, 53	syl5ibrcom 247	. . 3 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) ∧ 𝑑 ∈ 𝐷)) → ((𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽)) → 𝑑 = (𝑐 ∪ 𝑒)))
55	4	ffnd 6652	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑐 Fn (𝐼 ∖ 𝐽))
56	7	ffnd 6652	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑒 Fn 𝐽)
57		fnunres1 6593	. . . . . . . 8 ⊢ ((𝑐 Fn (𝐼 ∖ 𝐽) ∧ 𝑒 Fn 𝐽 ∧ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅) → ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) = 𝑐)
58	55, 56, 9, 57	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) = 𝑐)
59	58	eqcomd 2737	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)))
60		fnunres2 6594	. . . . . . . 8 ⊢ ((𝑐 Fn (𝐼 ∖ 𝐽) ∧ 𝑒 Fn 𝐽 ∧ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅) → ((𝑐 ∪ 𝑒) ↾ 𝐽) = 𝑒)
61	55, 56, 9, 60	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) ↾ 𝐽) = 𝑒)
62	61	eqcomd 2737	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽))
63	59, 62	jca 511	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽)))
64	63	adantrr 717	. . . 4 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) ∧ 𝑑 ∈ 𝐷)) → (𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽)))
65		reseq1 5921	. . . . . 6 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → (𝑑 ↾ (𝐼 ∖ 𝐽)) = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)))
66	65	eqeq2d 2742	. . . . 5 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → (𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ↔ 𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽))))
67		reseq1 5921	. . . . . 6 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → (𝑑 ↾ 𝐽) = ((𝑐 ∪ 𝑒) ↾ 𝐽))
68	67	eqeq2d 2742	. . . . 5 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → (𝑒 = (𝑑 ↾ 𝐽) ↔ 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽)))
69	66, 68	anbi12d 632	. . . 4 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → ((𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽)) ↔ (𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽))))
70	64, 69	syl5ibrcom 247	. . 3 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) ∧ 𝑑 ∈ 𝐷)) → (𝑑 = (𝑐 ∪ 𝑒) → (𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽))))
71	54, 70	impbid 212	. 2 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) ∧ 𝑑 ∈ 𝐷)) → ((𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽)) ↔ 𝑑 = (𝑐 ∪ 𝑒)))
72	1, 35, 39, 41, 71	f1o2d2 42274	1 ⊢ (𝜑 → 𝐻:(𝐶 × 𝐸)–1-1-onto→𝐷)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {crab 3395  Vcvv 3436   ∖ cdif 3894   ∪ cun 3895   ∩ cin 3896   ⊆ wss 3897  ∅c0 4280   class class class wbr 5089   × cxp 5612  ^◡ccnv 5613  dom cdm 5614   ↾ cres 5616   “ cima 5617  Rel wrel 5619   Fn wfn 6476  ⟶wf 6477  –1-1-onto→wf1o 6480  (class class class)co 7346   ∈ cmpo 7348   ↑_m cmap 8750  Fincfn 8869   finSupp cfsupp 9245  0cc0 11006  ℕcn 12125  ℕ₀cn0 12381  ℤcz 12468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-supp 8091  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-fsupp 9246  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-neg 11347  df-nn 12126  df-n0 12382  df-z 12469

This theorem is referenced by:  evlselv  42628