Theorem evlselvlem 41791

Description: Lemma for evlselv 41792. 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 21844	. . . . . 6 ⊢ (𝑐 ∈ 𝐶 → 𝑐:(𝐼 ∖ 𝐽)⟶ℕ0)
4	3	ad2antrl 727	. . . . 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 21844	. . . . . 6 ⊢ (𝑒 ∈ 𝐸 → 𝑒:𝐽⟶ℕ0)
7	6	ad2antll 728	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑒:𝐽⟶ℕ0)
8		disjdifr 4468	. . . . . 6 ⊢ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅
9	8	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅)
10	4, 7, 9	fun2d 6755	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒):((𝐼 ∖ 𝐽) ∪ 𝐽)⟶ℕ0)
11		evlselvlem.j	. . . . . . 7 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
12		undifr 4478	. . . . . . 7 ⊢ (𝐽 ⊆ 𝐼 ↔ ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
13	11, 12	sylib 217	. . . . . 6 ⊢ (𝜑 → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
14	13	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
15	14	feq2d 6702	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒):((𝐼 ∖ 𝐽) ∪ 𝐽)⟶ℕ0 ↔ (𝑐 ∪ 𝑒):𝐼⟶ℕ0))
16	10, 15	mpbid 231	. . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒):𝐼⟶ℕ0)
17		unexg 7745	. . . . . 6 ⊢ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) → (𝑐 ∪ 𝑒) ∈ V)
18	17	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒) ∈ V)
19		0zd 12594	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 0 ∈ ℤ)
20	10	ffund 6720	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → Fun (𝑐 ∪ 𝑒))
21	2	psrbagfsupp 21846	. . . . . . 7 ⊢ (𝑐 ∈ 𝐶 → 𝑐 finSupp 0)
22	21	ad2antrl 727	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑐 finSupp 0)
23	5	psrbagfsupp 21846	. . . . . . 7 ⊢ (𝑒 ∈ 𝐸 → 𝑒 finSupp 0)
24	23	ad2antll 728	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑒 finSupp 0)
25	22, 24	fsuppun 9404	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) supp 0) ∈ Fin)
26	18, 19, 20, 25	isfsuppd 9384	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒) finSupp 0)
27		fcdmnn0fsuppg 12555	. . . . 5 ⊢ (((𝑐 ∪ 𝑒) ∈ V ∧ (𝑐 ∪ 𝑒):((𝐼 ∖ 𝐽) ∪ 𝐽)⟶ℕ0) → ((𝑐 ∪ 𝑒) finSupp 0 ↔ (◡(𝑐 ∪ 𝑒) “ ℕ) ∈ Fin))
28	18, 10, 27	syl2anc 583	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) finSupp 0 ↔ (◡(𝑐 ∪ 𝑒) “ ℕ) ∈ Fin))
29	26, 28	mpbid 231	. . 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 21843	. . . 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 712	. 2 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → (𝑐 ∪ 𝑒) ∈ 𝐷)
36	30	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝐼 ∈ 𝑉)
37		difssd 4128	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (𝐼 ∖ 𝐽) ⊆ 𝐼)
38		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ 𝐷)
39	32, 2, 36, 37, 38	psrbagres 41748	. 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 41748	. 2 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → (𝑑 ↾ 𝐽) ∈ 𝐸)
42	32	psrbagf 21844	. . . . . . . 8 ⊢ (𝑑 ∈ 𝐷 → 𝑑:𝐼⟶ℕ0)
43	42	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑑:𝐼⟶ℕ0)
44	43	freld 6722	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → Rel 𝑑)
45	43	fdmd 6727	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → dom 𝑑 = 𝐼)
46	40, 12	sylib 217	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼)
47	45, 46	eqtr4d 2770	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → dom 𝑑 = ((𝐼 ∖ 𝐽) ∪ 𝐽))
48	8	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅)
49		reldisjun 6030	. . . . . 6 ⊢ ((Rel 𝑑 ∧ dom 𝑑 = ((𝐼 ∖ 𝐽) ∪ 𝐽) ∧ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅) → 𝑑 = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽)))
50	44, 47, 48, 49	syl3anc 1369	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐷) → 𝑑 = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽)))
51	50	adantrl 715	. . . 4 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) ∧ 𝑑 ∈ 𝐷)) → 𝑑 = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽)))
52		uneq12 4154	. . . . 5 ⊢ ((𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽)) → (𝑐 ∪ 𝑒) = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽)))
53	52	eqeq2d 2738	. . . 4 ⊢ ((𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽)) → (𝑑 = (𝑐 ∪ 𝑒) ↔ 𝑑 = ((𝑑 ↾ (𝐼 ∖ 𝐽)) ∪ (𝑑 ↾ 𝐽))))
54	51, 53	syl5ibrcom 246	. . 3 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) ∧ 𝑑 ∈ 𝐷)) → ((𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽)) → 𝑑 = (𝑐 ∪ 𝑒)))
55	4	ffnd 6717	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑐 Fn (𝐼 ∖ 𝐽))
56	7	ffnd 6717	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑒 Fn 𝐽)
57		fnunres1 6660	. . . . . . . 8 ⊢ ((𝑐 Fn (𝐼 ∖ 𝐽) ∧ 𝑒 Fn 𝐽 ∧ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅) → ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) = 𝑐)
58	55, 56, 9, 57	syl3anc 1369	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) = 𝑐)
59	58	eqcomd 2733	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → 𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)))
60		fnunres2 6661	. . . . . . . 8 ⊢ ((𝑐 Fn (𝐼 ∖ 𝐽) ∧ 𝑒 Fn 𝐽 ∧ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅) → ((𝑐 ∪ 𝑒) ↾ 𝐽) = 𝑒)
61	55, 56, 9, 60	syl3anc 1369	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸)) → ((𝑐 ∪ 𝑒) ↾ 𝐽) = 𝑒)
62	61	eqcomd 2733	. . . . . 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 716	. . . 4 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) ∧ 𝑑 ∈ 𝐷)) → (𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽)))
65		reseq1 5973	. . . . . 6 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → (𝑑 ↾ (𝐼 ∖ 𝐽)) = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)))
66	65	eqeq2d 2738	. . . . 5 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → (𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ↔ 𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽))))
67		reseq1 5973	. . . . . 6 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → (𝑑 ↾ 𝐽) = ((𝑐 ∪ 𝑒) ↾ 𝐽))
68	67	eqeq2d 2738	. . . . 5 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → (𝑒 = (𝑑 ↾ 𝐽) ↔ 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽)))
69	66, 68	anbi12d 630	. . . 4 ⊢ (𝑑 = (𝑐 ∪ 𝑒) → ((𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽)) ↔ (𝑐 = ((𝑐 ∪ 𝑒) ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = ((𝑐 ∪ 𝑒) ↾ 𝐽))))
70	64, 69	syl5ibrcom 246	. . 3 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐶 ∧ 𝑒 ∈ 𝐸) ∧ 𝑑 ∈ 𝐷)) → (𝑑 = (𝑐 ∪ 𝑒) → (𝑐 = (𝑑 ↾ (𝐼 ∖ 𝐽)) ∧ 𝑒 = (𝑑 ↾ 𝐽))))
71	54, 70	impbid 211	. 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 41696	1 ⊢ (𝜑 → 𝐻:(𝐶 × 𝐸)–1-1-onto→𝐷)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {crab 3427  Vcvv 3469   ∖ cdif 3941   ∪ cun 3942   ∩ cin 3943   ⊆ wss 3944  ∅c0 4318   class class class wbr 5142   × cxp 5670  ^◡ccnv 5671  dom cdm 5672   ↾ cres 5674   “ cima 5675  Rel wrel 5677   Fn wfn 6537  ⟶wf 6538  –1-1-onto→wf1o 6541  (class class class)co 7414   ∈ cmpo 7416   ↑_m cmap 8838  Fincfn 8957   finSupp cfsupp 9379  0cc0 11132  ℕcn 12236  ℕ₀cn0 12496  ℤcz 12582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fsupp 9380  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-neg 11471  df-nn 12237  df-n0 12497  df-z 12583

This theorem is referenced by:  evlselv  41792